Estimating the coherence of noise in mid-scale quantum systems
Pedro Figueroa-Romero Miha Papič Adrian Auer Inés de Vega




pedro.romero@meetiqm.com
IQM Quantum Computers, Georg-Brauchle-Ring 23-25, 80992 Munich, Germany


IQM Quantum Computers, Georg-Brauchle-Ring 23-25, 80992 Munich, Germany

 Department of Physics and Arnold Sommerfeld Center for Theoretical Physics,

Ludwig-Maximilians-Universität München, Theresienstr. 37, 80333 Munich, Germany



IQM Quantum Computers, Georg-Brauchle-Ring 23-25, 80992 Munich, Germany


IQM Quantum Computers, Georg-Brauchle-Ring 23-25, 80992 Munich, Germany

 Department of Physics and Arnold Sommerfeld Center for Theoretical Physics,

Ludwig-Maximilians-Universität München, Theresienstr. 37, 80333 Munich, Germany




§ ABSTRACT 
While the power of quantum computers is commonly acknowledged to rise exponentially, it is often overlooked that the complexity of quantum noise mechanisms generally grows much faster. In particular, quantifying whether the instructions on a quantum processor are close to being unitary has important consequences concerning error rates, e.g., for the confidence in their estimation, the ability to mitigate them efficiently, or their relation to fault-tolerance thresholds in error correction. However, the complexity of estimating the coherence, or unitarity, of noise generally scales exponentially in system size. Here, we obtain an upper bound on the average unitarity of Pauli noise and develop a protocol allowing us to estimate the average unitarity of operations in a digital quantum device efficiently and feasibly for mid-size quantum systems. We demonstrate our results through both experimental execution on , a 5-qubit superconducting quantum computer, and in simulation with up to 10 qubits, discussing the prospects for extending our technique to arbitrary scales.




The main obstacle for quantum computing undeniably remains the presence of noise, causing a multitude of errors that limit the usefulness of currently available quantum processors. That is to say, not all quantum noise is made equal, and ironically, figures of merit that are efficient to access in an experiment –such as the fidelity of a quantum operation– are often by definition insensitive to many of the details of such noise. It is thus imperative to have a plethora of protocols with certain practical desiderata that can probe different aspects of noise, and that can then, in turn, provide actionable information to existing methods to suppress, mitigate, or fully correct potential errors.

Quantum noise can be described as an undesirable quantum transformation mapping the ideal noiseless output state of a quantum computation into some other valid, albeit unexpected, output quantum state. Focusing on the regime where such noise is effectively Markovian, i.e., it only depends on the input state and no other external context variables, a comprehensive classification of all possible errors can be made <cit.>, but in particular it is extremely important to be able to distinguish between coherent and incoherent error contributions.

Coherent noise can be described as a deterministic, undesired unitary transformation, e.g., it can arise due to imperfect control or calibration of quantum gates, and also due to certain types of crosstalk <cit.>. Incoherent or stochastic noise, on the other hand, is described by non-unitary transformations and arises purely statistically, e.g., a bit-flip occurring with a certain non-zero probability, or so-called depolarizing noise, whereby a state gets maximally mixed with a certain probability but remains intact with the complementary probability. Generally, Markovian quantum noise will be described by a quantum operation that contains both coherent and stochastic noise contributions. The reasons why distinguishing between these is important, include:

    
  * Coherent errors can, for example, be effectively suppressed by good experimental quantum control, while incoherent errors in isolation can be efficiently mitigated <cit.> and fully corrected <cit.>.
    
  * Coherent errors can accumulate in a much more detrimental way and provide error-rate estimates, such as average gate-fidelity, that can differ by orders of magnitude from worst-case estimates (i.e., fault-tolerance thresholds) <cit.>.
    
  * The interpretation of average gate fidelity as a reliable and meaningful error rate depends heavily on the noise having a low coherent contribution <cit.>.


The average coherence of a noisy quantum gate can be understood as a measurement of the rate at which noise shrinks the n-dimensional Bloch ball, as depicted in Fig. <ref>, and can be measured by its unitarity, which in turn can be estimated operationally by urb <cit.>. Despite being efficient and robust to spam errors, urb is not scalable in system size, as it requires estimating an exponential amount of expectation values within a rb-like protocol. The reason for this is not urb per se, but rather that fundamentally, the unitarity is a second-order functional of the noise, akin to the case of the so-called purity for a quantum state.



In general, the complexity of characterizing quantum noise comprehensively typically scales exponentially with system size <cit.>. Even the most efficient of protocols estimating a single figure of merit, such as those within the family of rb <cit.>, fail to do so at scale. However, with the quantum industry inevitably moving towards larger systems, there is a pressing need to find ways to generalize such tools to larger systems.

Recently, scalable rb-based techniques such as mrb <cit.> and birb <cit.>, among others <cit.>, have been developed, which can estimate the average fidelity of sets of quantum operations in a scalable, efficient and spam-robust way, for a large class of gate sets. On the other hand, rm techniques, and so-called classical shadows, have enabled the efficient estimation of properties of many-body quantum systems <cit.> (including state fidelity and purity), linear properties of gate sets <cit.>, and characterization of processes with memory <cit.>.

We harness the results of <cit.>, together with a scalable rb-inspired protocol, to develop a standalone technique allowing to simultaneously estimate the average fidelity and average unitarity of layers of operations in mid-scale quantum systems. Moreover, we first identify an interval for the unitarity of so-called stochastic Pauli noise in terms of average fidelity, enabling certification of whether average noise has a low coherent contribution and establishing meaningful error budgets via the average unitarity.

In  <ref> we introduce the main technical background, while in  <ref> we present a new upper bound on the unitarity of Pauli noise. In  <ref> we summarize existing results regarding rms, before presenting our protocol in  <ref>, which in  <ref> we demonstrate with experimental execution on  <cit.>. In the remaining  <ref> we discuss the bottlenecks for a scalable estimation of average coherence of noise, and finally, we draw some conclusions and prospects to overcome mid-scalability in  <ref>.



§ AVERAGE COHERENCE OF NOISE AND ITS RELATION WITH PURITY AND FIDELITY

It is well known that no physical system can be perfectly isolated. In particular, the loss of coherence of quantum states is one of the fundamental practical problems facing any quantum technology. In general, the dissipation of information to an external environment results in an increasingly mixed classical probability distribution over a set of possible quantum states. Such mixedness or uncertainty can be quantified by the purity of the respective density matrix ρ, given by (ρ^2). The purity takes extremal values of 1 for a pure state, and 2^-n for a maximally mixed state on n-qubits, and it can also be written as the Schatten 2-norm ρ_2^2, where X_2=[(XX^)]^1/2.

Logical operations or gates G in a quantum computer are described by unitary operators, which by definition preserve the purity of quantum states, i.e., for G(ρ_𝗂𝗇)=ρ_𝗈𝗎𝗍, we have (ρ_𝗂𝗇^2)=(ρ_𝗈𝗎𝗍^2). The corresponding, real noisy operation, G_𝗇𝗈𝗂𝗌𝗒, however, is described by a more general cp map, such that for G_𝗇𝗈𝗂𝗌𝗒(ρ_𝗂𝗇)=ρ_𝗈𝗎𝗍, we have (ρ_𝗈𝗎𝗍^2)≤(ρ_𝗂𝗇^2). The reduction in purity of states when acted on by a noisy channel is related to the unitarity of such channel, a measure of “how far” G_𝗇𝗈𝗂𝗌𝗒 is from being unitary. Because locally we can always relate a noisy channel with an ideal gate by G_𝗇𝗈𝗂𝗌𝗒:=E∘G (<cit.>) for some cp map E, the unitarity of the noisy gate equivalently measures how far E is from being unitary.

The average unitarity of a quantum channel can be defined as in <cit.>, by

    (E)    := 2^n2^n-1_ψ∼𝖧𝖺𝖺𝗋E^'(ψ)_2^2,

where E^'(·) := E( ·-1/2^n), with 1 being an n-qubit identity operator, and the normalization 2^n/(2^n-1) ensuring that 0≤(E)≤1, and with (E)=1 if and only if E —and hence the noisy operation G_𝗇𝗈𝗂𝗌𝗒— is unitary. Since the purity of a noisy output state may be written in the form (ρ_𝗈𝗎𝗍^2)=E(ρ_𝗂𝗇)_2^2, the definition in Eq. (<ref>) implies that the average purity of a noisy state is, in general, a combination of average unitarity and trace-decrease and/or non-unital terms [ See, e.g., Eq. (<ref>) for an explicit expression.].

The unitarity only quantifies how unitary G_𝗇𝗈𝗂𝗌𝗒 is, but not whether it is the target unitary, i.e., it does not measure whether G_𝗇𝗈𝗂𝗌𝗒 is far from the ideal G. A figure of merit aiming to quantify this distinction is the average gate fidelity,

    𝖥(G_𝗇𝗈𝗂𝗌𝗒,G)    := _ψ∼𝖧𝖺𝖺𝗋[G_𝗇𝗈𝗂𝗌𝗒(ψ)G(ψ)] 
       = _ψ∼𝖧𝖺𝖺𝗋⟨ψ|E(ψ)|ψ⟩ := 𝖥(E),

where implicitly we denote 𝖥(E):=𝖥(E,I), the average gate fidelity of E with respect to the identity I (and in a slight abuse of notation, ψ=|ψ⟩⟨ψ|). The average gate fidelity satisfies 1/(2^n+1)≤𝖥(E)≤1, with 𝖥(E)=1 if and only if E=I (equivalently if and only if G_𝗇𝗈𝗂𝗌𝗒=G). With respect of , in <cit.> it is shown that the average unitarity satisfies (E) ≥f(E)^2, where

    f(E) = 2^n(E)-12^n-1,

is the so-called average polarization of E, and with saturation occurring for a depolarizing (purely incoherent) channel.

The unitarity can also be written in terms of fidelity as (E) = (2^n/(2^n-1))𝖥(E^' E^'), where E^' is the adjoint map of E^'[ The adjoint map Φ^ of a cp map Φ is defined by [AΦ(B)]=[Φ^(A)B], or equivalently by conjugating the Kraus operators of Φ.]. Thus, the average unitarity can also be understood as a type of second-order function of the fidelity of E^', quantifying on average how distinguishable is the composition E^' E^' from the identity. Under the assumptions that E approximately models the average noise of a gate set {G_i} in a temporally uncorrelated (Markovian), time-independent and gate-independent way, 𝖥(E) can be estimated through rb, and the unitarity (E) can be estimated through urb <cit.>. Neither technique is scalable in the number of qubits, however, recently the mrb and birb variants were developed in <cit.> allowing efficient and scalable estimation of at least 𝖥(E) for a large number of qubits.



§ PAULI NOISE UNITARITY

Incoherent noise is tp and unital, and a distinction from coherent noise is that it generates no net rotation of the state space <cit.>. Pauli noise is a special case of incoherent noise described by a channel ρ↦∑_P∈𝖯𝖺𝗎𝗅𝗂_nα_PPρP, where α_P is the Pauli error probability associated to a n-qubit Pauli operator P. Its relevance stems not only from that of the Pauli basis in quantum information theory and experiment but also from its extensive application in quantum error characterization <cit.>, quantum error mitigation <cit.>, quantum error correction <cit.> and in its connection with fault tolerance thresholds <cit.>. Any quantum channel E can be averaged to a corresponding Pauli channel E^ by means of its Pauli twirl, E↦E^(·):=∑_P∈𝖯𝖺𝗎𝗅𝗂_n4^-nPE(P·P)P, i.e., reduced to a Pauli channel with the same Pauli error probabilities as E.

[Fidelity bound for Pauli unitarity] The unitarity of the Pauli-twirled channel E^ of E is bounded by

    f(E)^2 ≤(E^) ≤4^n-2(2^n-1)^2 r(E)^2+f(E)^2,

where f is average polarization defined by Eq. (<ref>), and r:=1- is average infidelity.


The proof is detailed in in Appendix <ref>. Given an average layer (in)fidelity, the bound in Ineq. (<ref>) allows us to estimate whether average noise is close to Pauli through the average unitarity. Furthermore, the Pauli-twirled channel E^ of a given noise channel E can be efficiently approximated operationally by rc <cit.>, so Ineq. (<ref>) enables to certify rc through the average unitarity. Generally, for small average infidelity, Ineq. (<ref>) will be tight, and the average unitarity will need to approach the square of the average polarization, f(E)^2, for the average noise to be strictly Pauli. This can be seen more clearly for a fixed number of qubits n, as in Fig. <ref>.

The upper bound in Ineq. (<ref>) overestimates the average unitarity of Pauli noise by a proportion of cross-products of non-identity Pauli error rates, (∑α_P)^2-∑α_P^2, and further assumes tp noise, so while a unitarity outside the Pauli bounds guarantees that average noise will contain coherent error contributions, a unitarity value within the Pauli interval (<ref>) solely points to the likelihood of the average noise being Pauli.





§ PURITY AND FIDELITY VIA RANDOMIZED MEASUREMENTS

The main roadblock for a scalable urb stems from purity being a quadratic function of a quantum state, in general requiring knowledge of all its components, e.g., as described in <cit.> requiring state tomography within a rb protocol. The efficient estimation of purity, however, was one of the first problems giving rise to the rm and shadow tomography techniques <cit.>. While for the particular case of purity, the number of required measurements for a given accuracy still scales exponentially, the exponent for rms is significantly smaller than for full tomography <cit.>.

We will focus on the result of <cit.>,

    (ρ^2 ) = 2^n∑_𝐬,𝐬^' (-2)^-h(𝐬,𝐬^')_𝖴_i∼𝖧𝖺𝖺𝗋𝖯_𝖴^(𝐬)𝖯_𝖴^(𝐬^'),

where here 𝖴=_i=1^n𝖴_i is a random unitary, with _𝖴_i∼𝖧𝖺𝖺𝗋 denoting averaging with the Haar measure over the each 𝖴_i, 𝐬 and 𝐬^' are n-bit strings, h(𝐬,𝐬^') is the Hamming distance (number of distinct bits) between them, and the

    𝖯_𝖴^(𝐬) := ⟨𝐬|𝖴^ρ 𝖴|𝐬⟩,

are probabilities of observing the n-bit string 𝐬 upon randomizing with 𝖴, and which are estimated in experiment. Since Eq. (<ref>) involves two copies of 𝖴 and 𝖴^, it suffices that the local unitaries 𝖴_i belong to a unitary 2-design <cit.>, such as the uniformly-distributed Clifford group.

By a rm, here we will mean a random unitary operation 𝖴, followed by a projective measurement in the computational basis, 𝖴|𝐬⟩ [ More strictly, this corresponds to a rm element, and the concept of a rm can be formalized with Random povms, as in <cit.>]. In Eq. (<ref>), this can equivalently be read as randomizing the state ρ with 𝖵=𝖴^, and then performing a computational basis measurement.

Purity can alternatively be estimated through the so-called shadow of the state, constructed via rms. That is, one can equivalently construct proxy states ρ̂_𝗌 through the probabilities 𝖯_𝖴(𝗌) and compute the purity through products (ρ̂_𝗌ρ̂_𝗌^') <cit.>. This is an equivalent approach with similar performance guarantees <cit.>; here we employ Eq. (<ref>) because it can be used within a rb-like protocol straightforwardly.

Operationally, the purity is truly important once in light of an associated fidelity <cit.>: notice that the probabilities 𝖯_𝖴^(𝐬) can be recycled in a straightforward way to estimate the fidelity of ρ with respect to an ideal pure state |ψ⟩, through a slight modification to Eq. (<ref>), as

    ⟨ψ|ρ|ψ⟩ = 2^n∑_𝐬,𝐬^' (-2)^-h(𝐬,𝐬^')_𝖴_i∼𝖧𝖺𝖺𝗋𝖯_𝖴^(𝐬)𝖰_𝖴^(𝐬^'),

where here now 𝖰_𝖴^(𝐬)=|⟨ψ|𝖴|𝐬⟩|^2 is the probability of observing the n-bit string 𝐬 upon randomizing with 𝖴=_i𝖴_i and measuring |ψ⟩. This comes at the expense of estimating the ideal probabilities 𝖰_𝖴^(𝐬), but this can be done classically in an efficient way since the 𝖴_i can be replaced by single-qubit Clifford unitaries.



§ AVERAGE UNITARITY THROUGH RANDOMIZED MEASUREMENT CORRELATIONS

While we do not require the same setup as in mrb or birb, we mainly follow the framework and notation of <cit.>. We will, in particular, only consider single-qubit and two-qubit gate sets, G_1 and G_2, distributed according to probability distributions Ω_1 and Ω_2, respectively. We will refer to instructions with these gate sets on n-qubits, simply as layers, and denote the corresponding set of layers by L(G), where G=G_1∪G_2. In practice, a rule on how to apply the sampled gates must also be specified, depending e.g., on the topology, connectivity, and the desired density of the gates.

We construct n-qubit random circuits, of layer circuit circuit depth m, of the form

    𝖢_m=𝖫_m𝖫_m-1⋯𝖫_1,

where the right-hand side is read (or acts) from right to left, and where each 𝖫_i=𝖫_i^(2)𝖫_i^(1) is a composite layer, made up of subsequent applications of a layer 𝖫_i^(1)∈L(G_1), of only parallel single-qubit gates, and a layer 𝖫_i^(2)∈L(G_2), of only parallel two-qubit gates. We will say that all 𝖫_i∈L(G) are random layers distributed according to Ω, which is such that Ω(𝖫_i)=Ω_1(𝖫_i^(1))Ω_2(𝖫_i^(2)), and we refer to 𝖢_m in Eq. (<ref>) as a Ω-distributed random circuit of depth m.



We use the term depth to mean the number of layers entering the given Ω-distributed circuit, before transpilation to a given basis gate set. We illustrate these concepts in Fig. <ref>. We also point out that while the choice of the pair (G,Ω) is a priori arbitrary, here we will only consider G_1:=𝖢𝗅𝗂𝖿_1 being the Clifford group (or effectively any other efficient single-qubit unitary 2-design) and Ω_1 a uniform distribution.

Finally, below we distinguish estimators of expectation values of a random variable by a circumflex, i.e., 𝖷̂ represents the estimator of the expectation of some random variable 𝖷.



 §.§ The protocol

With this setup, the protocol consists of:

    
  *  Preparation: Generate N_𝖢_m^* samples of n-qubit Ω-distributed circuits 𝖢_m for all given circuit depths m, each with a different layer 𝖵∈L(𝖢𝗅𝗂𝖿_1) sampled uniformly, preppended to it. We will denote 𝖢^*_m:=𝖢_m𝖵.
    
    
  *  Quantum Execution: Estimate probabilities
    
    𝖯_𝖶,𝖢_m^*^(𝐬) := ⟨𝐬|𝖶 𝖢_m𝖵(|0⟩⟨0|)𝖵^𝖢_m^†𝖶^†|𝐬⟩,

    of observing the n-bit string 𝐬, for all circuit depths m, all N_𝖢_m^* circuit samples and a given number N_𝖶 of layer samples 𝖶∈L(G_1), sampled according to Ω_1. We denote the corresponding estimator quantities, given a finite number of measurements N_𝗆𝖾𝖺𝗌 by
    
    𝖯̂_𝖶,𝖢_m^*^(𝐬)=1N_𝗆𝖾𝖺𝗌∑_i=1^N_𝗆𝖾𝖺𝗌1_𝗌(𝐱_i),

    where 𝐱_i is a random variable describing the ith projective measurement of the n-qubits in the computational basis, with 1_𝐬(𝐱)=1 if 𝐱=𝐬 or 0 otherwise.
    
    
  *  Classical post-processing: Estimate the average purity tr(ρ_m^2) for all depth m states ρ_m:=𝖢_m𝖵(|0⟩⟨0|)𝖵^𝖢_m^†, via
    
    𝔓̂_m = 2^n∑_𝐬,𝐬^'
    𝖶,𝖢_m^*(-2)^-h(𝐬,𝐬^')𝖯̂^(𝐬)_𝖶,𝖢_m^*𝖯̂^(𝐬^')_𝖶,𝖢_m^*N_𝖶N_𝖢_m^*,

    where h(𝐬,𝐬) is the number of distinct elements between the n-bit strings 𝐬 and 𝐬^'. Estimate the probabilities 𝖰̂^(𝐬)_𝖶,𝖢_m^* := |⟨𝐬|𝖶^(𝗂𝖽)|ψ^(𝗂𝖽)_m⟩|^2 for all noiseless states, ψ^(𝗂𝖽)_m:=𝖢_m^(𝗂𝖽)𝖵^(𝗂𝖽)|0⟩ of depth m —with the label (𝗂𝖽) here denoting noiseless quantities—, and the corresponding average fidelities ⟨ψ_m|ρ_m|ψ_m⟩, via

    
    𝔉̂_m = 2^n∑_𝐬,𝐬^'
    𝖶,𝖢_m^*(-2)^-h(𝐬,𝐬^')𝖯̂^(𝐬)_𝖶,𝖢_m^*𝖰̂^(𝐬^')_𝖶,𝖢_m^*N_𝖶N_𝖢_m^*.

    
    Both estimators can be rendered unbiased and improved by other considerations as in  <ref>.


Whenever the Ω-distributed circuits are generated by layers containing non-Clifford gates in a way that becomes impractical to simulate (e.g., whereby there is a high density of such non-Cliffords or for a large number of qubits), the average layer fidelity can alternatively be estimated through mrb <cit.>; the trade-off is having to construct and measure corresponding mirror circuits. The estimations of Protocol <ref> and mrb by definition will agree upon employing the same ensemble of layers; in Appendix <ref> we numerically observe such agreement.


[Exponential decay]Throughout Appendices <ref>, <ref> and <ref> we show that, under the circumstances detailed below,

    _Ωtr(ρ_m^2) ≈ A (E)^m + 12^n,

where 0≤A≤1, and 

    𝗎(E)    = 2^n(E^E)-12^n-1 = f(E^E)

is the average layer unitarity of the noise, where E:=L_i_𝗇𝗈𝗂𝗌𝗒L_i^ is approximately the noise channel corresponding to any layer 𝖫_i with associated map L_i(·):=𝖫_i(·)𝖫_i^, and (E^E) is the average fidelity of E^E with respect to the identity.

Similarly, the decay in average fidelity can be seen to correspond to

    _Ω⟨ψ_m|ρ_m|ψ_m⟩≈αf(E)^m + 12^n,

where 0≤α≤1, similar to a usual rb decay for unital spam.

In such case, the estimators in Eq. (<ref>) and Eq. (<ref>) of protocol <ref>, can be fit to the respective decays in depth m, whereby both average layer unitarity and fidelity can be estimated.




  §.§.§ Conditions for an exponential decay

While a general functional form in terms of any Markovian noise model for the average sequence purity and fidelity can be obtained, as in Appendices <ref>, <ref> and <ref>, ensuring that it will follow a simple exponential decay as in Eq. (<ref>) relies on the noise satisfying certain approximate conditions and the pair (G,Ω) having certain properties, similar to any rb-based technique. Aside from standard assumptions such as Markovianity (i.e., that noise is not temporally correlated), time-independence and weak gate-dependence [ Here meaning with time- and gate-dependent contributions being negligible on average; see e.g., <cit.>.], Result <ref> requires the following:


    
  *  That noise is approximately trace-preserving and unital, i.e., without significant leakage or entropy-decreasing contributions.
    
    
  *  That the Ω-distributed circuits approximate a unitary 2-design, to the effect that _𝖫∼ΩL^XL(·)≈p(·)+(1-p)1/2^n, where p≤1 only dependent on X; i.e., the average composition of noisy layers is approximately equivalent to a depolarizing channel encoding some information about X. This is formalized and discussed in Appendix <ref>.


Assumption <ref> implies that the purity decay only depends on the unitarity of the average noise, and not on its non-unital or trace-decreasing contribution. Otherwise, if this condition is not satisfied, the average unitarity will not be described by Eq. (<ref>), but rather the most general Eq. (<ref>), and the purity decay will be a convex combination of average unitarity and trace-decreasing contributions of the average noise, as expressed by Eq. (<ref>) of Appendix <ref> (in agreement with <cit.>). The implications for the fidelity estimation in this case are similar, since the polarization factor would not be described by Eq. (<ref>) but rather a more general f_t(E)=(2^n(E)-t/(2^n-1) for t=[E(1/2^n)] quantifying how non-tp the channel is <cit.>. In other words, the main implication is that the fitting procedure would generally be slightly more complicated for non-unital, non-tp noise.

Assumption <ref> implies that we can identify 𝗎(E)=p^2 as given by Eq. (<ref>), i.e., the average unitarity is simply equal to the polarization of the average composition of noisy layers. This assumption is slightly stronger than that in <cit.>, and generally means that up to the second moment and a small positive ϵ, the Ω-distributed circuits we employ should reproduce the statistics of the uniform Haar measure on the unitary group on the n qubits. While the choice of (G_1,Ω_1) being the uniform single-qubit Clifford group already generates separately a unitary 2-design on each qubit (projecting noise to a Pauli channel <cit.>), the choice (G_2,Ω_2) should be such that the random Ω-distributed circuits are highly scrambling, as defined in <cit.>, which can be achieved by it containing at least an entangling gate with high probability. While this does not ensure that the circuits will approximate a unitary 2-design in an increasing number of qubits, it suffices in practice for mid-scale systems. This is discussed in detail in Appendix <ref>.



 §.§ Mid-scale

In  <ref>, we will discuss the reasons why Protocol <ref> is practical and feasible for at least 10-qubits and generally within tenths of qubits, which is what we refer to as mid-scale. This is in the sense that all classical aspects, i.e., steps <ref> and <ref> can be managed with a standard current-technology laptop and with the quantum execution requiring a mild number of circuits, rms, and measurement shot samples, as exemplified in  <ref>.



 §.§ Unbiased and Median of Means estimators

There are at least two simple but effective ways in which reliable results can be ensured through the estimators in Eq. (<ref>) and Eq. (<ref>): i) using unbiased estimators [ An estimator of some statistical parameter is said to be unbiased or faithful when its expectation matches the real expected value of the parameter, or otherwise it is said to be biased.] and ii) using moms estimators.

In <cit.> it is pointed out that, even though the probability estimator in Eq. (<ref>) is faithful, i.e. [𝖯̂]=𝖯 (where we have dropped all indices), it is biased for any other positive integer power. A unique unbiased estimator can nevertheless be built for any power, and in particular, 𝖯̂_2:=𝖯̂(𝖯̂N_𝗆𝖾𝖺𝗌-1)/(N_𝗆𝖾𝖺𝗌-1) is an unbiased estimator of 𝖯^2. Thus all terms in Eq. (<ref>) and Eq. (<ref>) for the n-bit string 𝐬=𝐬^' should be computed through 𝖯̂_2 to render average sequence fidelity and purity estimations unbiased.

On the other hand, a simple but highly effective way to reduce the uncertainty associated to a mean estimator is to use moms estimators: given K samples of estimators 𝔉̂_m^(1),⋯,𝔉̂_m^(K) for the average fidelity of circuits of depth m in Eq. (<ref>), the moms estimator of such samples is 𝔉̂_m^𝖬𝗈𝖬𝗌(K):=𝗆𝖾𝖽𝗂𝖺𝗇(𝔉̂_m^(1),⋯,𝔉̂_m^(K)), or similarly for the case of the average purity in Eq. (<ref>). In total, this requires N_𝖶N_𝖢_m^*K circuit samples to be measured N_𝗆𝖾𝖺𝗌 times, which however, is generally more robust compared with simply constructing an empirical estimator of the mean with the same amount of samples <cit.>. While it has been noted that when measurements are randomized with the single-qubit Clifford group both types of estimators converge similarly to the true mean <cit.>, moms has the concentration property of the probability of observing outliers from the true mean decreasing exponentially in K <cit.>.



§ EXPERIMENT ON IQM SPARK (TM)

We now demonstrate the execution of Protocol <ref> through both an experiment on  <cit.>, a commercial 5-qubit superconducting quantum system by IQM targeting education in quantum computing; technical details of the hardware can be seen in <cit.>. The five qubits in the  are connected in a star-shape, with a central qubit connected to the four remaining qubits. The experiments were performed at specific dates, pointed out where the respective results are displayed, and only reflect the performance of the hardware at such point in time.



We generated Ω-distributed circuits using the gate set G=𝖢𝗅𝗂𝖿_1∪{𝖢𝖹}, with 𝖢𝗅𝗂𝖿_1 being the uniformly-distributed single-qubit Clifford group (i.e., Ω_1 assigning 1/24 probability for each Clifford and Ω_2 probability 1 of sampling 𝖢𝖹). The sampling of layers was done employing the edge-grab sampler, defined in <cit.>, with a two-qubit gate density (the expected proportion of qubits occupied by two-qubit gates) of 1/2. The edge-grab sampler considers layers with two-qubit gates only on connected qubits, and where a single logical layer consists of a parallel mixture of one- and two-qubit gates (sampled according to Ω), i.e., there is no more than one gate acting on a given qubit in any layer.
 
The native set of gates is {r_θ,φ,𝖢𝖹}, where r_θ,φ is a single-qubit rotation of angle θ around the cos(φ)X + sin(φ)Y axis, 𝖢𝖹 is a controlled-Z two-qubit gate. The 𝗆𝖾𝖺𝗌𝗎𝗋𝖾𝗆𝖾𝗇𝗍 operator is a computational basis projective measurement operation, and a 𝖻𝖺𝗋𝗋𝗂𝖾𝗋 object is used to prevent the layers defined through 𝔾 from being compiled together. The definition of depth that we adopt is that of the number of layers defined through 𝔾, i.e., before transpilation to the native gate set.

We extracted decay rates of the average state fidelity and purity in increasing layer depths according to Protocol <ref>, and thus layer fidelity and unitarity, for Ω-distributed circuits in arbitrary combinations of n=1,2,…,5 qubits. In both cases, the sample parameters we employ are N_𝖢_m^*=12 random Ω-distributed circuits, times N_𝖶=10 randomized measurement layers, and times N_𝗆𝖾𝖺𝗌=2^11 shots per measurement; we furthermore employ unbiased estimators for square probabilities and either K=1 for n=1,2,3 or K=2 for n=4,5 momsmomsmoms estimators.



In Fig. <ref> we show the purity decays in increasing circuit depth for each number of qubits, determined via the estimators in Eq. (<ref>), where, by fitting the decay model in Eq. (<ref>) to the averages, we estimate the corresponding average unitarity. It is worth noticing that we used no median of means for up to three qubits, and thus the individual smaller points in the plot do not represent purities as these have not been averaged over all circuit samples; this is a reason why the violin distributions can stretch beyond 1. Nevertheless, it is expected for the spread in individual outputs to be larger for larger values of purity. Importantly, while averages (larger points) do not all fall exactly on the respective exponential (particularly for n=1), deviations do not appear alarmingly high. Finally, spam contributions can be seen to lead to a shift in the offset of the curve for n=3.

It is important to notice that in Fig. <ref>, the unitarity does not appear to be monotonically decreasing in qubit count. In this case, however, this apparent increase could also be explained away by the uncertainty of the data and could be at most a plateau in unitarity; a way to decrease the uncertainties could be by increasing the number of moms. This is nevertheless an interesting feature and one that could be investigated further also in the context of determining crosstalk or other types of correlations at the unitarity level.



Following this, in Fig. <ref> we show the respective fidelity decays, computed together with the measurement counts from the same experiment and the estimation of the corresponding noiseless probabilities, according to estimator in Eq. (<ref>). Similarly here, the distributions tend to show a larger spread for smaller n, where only the averages represent the corresponding fidelity estimation. While the decays display different spam contributions, the averages do decrease in system size; the uncertainties could similarly be reduced e.g., by increasing the amount of moms. In Appendix <ref> we compare with mrb results gathered weeks before, showing qualitative agreement despite a clear difference in the respective distributions. Since error bars remain relatively large for fidelities, and hence for the estimated Pauli unitarity intervals, an alternative could be to construct mirror circuits (without the rm Clifford gates appended) within the same experiment and then instead estimate fidelities via mrb for smaller uncertainties.

This finally enables us to get an overall picture of both average layer fidelities and unitarities, together with an estimation of whether noise falls within the Pauli unitarity bound in Eq. (<ref>) or it contains a larger coherent contribution. In Fig. <ref> we plot together the average outputs retrieved in Fig. <ref> and Fig. <ref>, together with estimations of Pauli unitarity bounds given the average fidelity estimates, computed via Eq. (<ref>). Fig. <ref> gives an overall picture of a coherent noise budget, which could in turn inform the overhead of Pauli twirling techniques. It is relevant to notice that both it is possible for the average unitarity to increase even without there being necessarily correlations between qubits, and that generally the coherence of noise could remain relatively large with respect to fidelity for larger systems.



§ SCALING BOTTLENECKS



 §.§ Sample Complexity

The main roadblock for estimating the average coherence of noise in large systems (or even just the purity of quantum states) with rms, is the required number of measurements to have an estimation within a given error, i.e., its sample complexity, which is then by definition manifested by the variance associated to such rms. In Protocol <ref>, statistical uncertainty stems from the sampling of the Ω-distributed circuits, that of the rm elements, and the number of measurements per rm element; these quantities correspond to N_𝖢_m^* and N_𝖶 in the estimators of Eq. (<ref>) and Eq. (<ref>), and to N_𝗆𝖾𝖺𝗌 of Eq. (<ref>), respectively.

The main limitation to scalability is imposed by N_𝗆𝖾𝖺𝗌: using local rms via Eq. (<ref>) to estimate purity to precision 1/√(N_𝖶), it was observed numerically in <cit.> that it scales approximately as N_𝗆𝖾𝖺𝗌∼2^0.75n in number of qubits n. Generally, within the classical shadows framework, the required number of measurements can be seen to scale exponentially in a similar fraction of n and linearly in the actual purity of the state <cit.>.

Mixedness reduces the sample complexity, as also observed numerically in <cit.>, and with Ω-distributed circuits, quantum states will naturally get increasingly mixed in increasing depth. Coincidentally, thus, a high percentage of coherent error contributes to a higher variance not only in average fidelity estimation but also in average unitarity estimation itself.

When it comes to Ω-distributed circuit samples when estimating average fidelity, the variance associated with it can be directly linked to the average amount of coherent noise <cit.>. How to control this uncertainty, as well as guarantees when choosing the number of sample circuits and number of measurements, has been widely studied for Clifford rb <cit.>; furthermore, <cit.> obtained a bound on the variance for the purity in circuits within (an improved version of) Clifford urb, assuming unital noise. While we are not aware of analogous bounds on the variance for scalable rb (namely employing Ω-distributed circuits), it has been observed numerically that similar behavior follows (at least when layers are made up only of either generators-of or Clifford gates themselves) <cit.>, and that Pauli twirling <cit.> generally suppresses the variance of expectation values under any sequence of operations, regardless of whether noise is correlated (either spatially or temporally) or not <cit.>.

A prospect could therefore be to benchmark average Pauli-dressed gates, aiming to obtain a unitarity within Ineq. (<ref>), with the trade-off being to sample an extra number of Pauli layers per circuit. Here we do not attempt to derive a rigorous bound on the variance for the average layer unitarity, but as we have demonstrated experimentally in  <ref>, while for a given total number of samples the uncertainty is larger for the unitarity than for fidelity, estimating it nevertheless remains tractable for mid-scale systems.



 §.§ Practical bottlenecks

There are two other practical considerations to take into account to estimate at which scale Protocol <ref> remains feasible: the spam factors in the exponential decay in Eqs. (<ref>),(<ref>), and the number of terms entering in the sum of Eqs. (<ref>),(<ref>).

While our technique gives spam-independent estimates of the average unitarity, it can nevertheless make the exponential decays drop way too quickly, so that fitting becomes unfeasible. This is a shared problem of all rb-based techniques, and other than simply improving readouts and/or state preparation, there could be ways of removing the 2^-n offset, for example, but the multiplicative factor is still determined by the unitarity of the spam [ For example, if we assume that spam noise (i.e. that which we would attribute to the initial and final single-qubit-only layers) is global depolarizing (as done for mrb <cit.>) with polarization p, then the multiplicative factor in Eq. (<ref>) is A≲p^4.]. More than a rb problem this might be an issue in general when trying to estimate global, high-weight properties on large systems, essentially because signal readouts eventually become too small.

On the other hand, while Eq. (<ref>) and Eq. (<ref>) are remarkable mathematically, in practice they involve a sum over all possible pairs of n-bit strings. Naively, this means summing up 4^n probability terms; for the case of the purity, this can be reduced at least to 2^n-1(2^n-1), given that the Hamming distance is symmetric. This computation can be done easily on a classical machine in parallel, but it can also become rather unpractical (as it furthermore would need to be done for every single purity estimate). While this is a practical problem, it is fundamentally tied to the fact that the purity of a state involves all the elements of the density matrix.



§ CONCLUSIONS

We have derived an upper bound on the average unitarity of Pauli-twirled noise solely in terms of the average fidelity of the respective bare noise, and we have established a protocol enabling the estimation of the average unitarity for a broad class of quantum circuits in digital systems, independent of technology platform or architecture, with tenths of qubits. Our results have been inspired both by novel scalable rb techniques, as well as rm methods. We have shown that reliable estimation of both average operation coherence and fidelity in such systems can be done under mild conditions and with a reasonable sampling overhead. Finally, we demonstrated our results in experiment on up to 5 qubits and numerically in simulation with up to 10 qubits.

A fully scalable estimation of the average coherence of noise will almost surely rely on modularity, e.g., by using smaller-scale unitarity information extracted from smaller qubit subsets and then combined with a Simultaneous rb-like protocol.

A promising way of moving forward in this direction is that of <cit.>, where the authors establish a method to estimate entropy and entanglement of a quantum state in polynomially-many measurements, relying on conditions essentially satisfied when spatial correlation lengths in the system remain finite and for systems in a one-dimensional topology. Their main result suggests that generally one could partition any large n-qubit system into smaller subsystems and reconstruct the global n-qubit purity from the local purity of pairs of contiguous subsystems, like stitching together the bigger purity with a number of measurements that overall grow polynomially in n. As a direct consequence, if such results hold more generally, our protocol could be used to estimate global average noise coherence with a similar sample complexity.

Nevertheless, several things remain unclear, e.g., what classes of (Markovian) noise such a method would hold for, how it could be phrased beyond one-dimensional qubit arrays, or how one would certify the global purity estimations, among others. As we have argued, however, the average coherence of noise is a crucial figure of merit for benchmarking the error mechanisms limiting the performance of quantum computers, perhaps only standing after average layer fidelity in terms of importance as a benchmark, so the aforementioned hurdles must soon be overcome.


The authors acknowledge support from the German Federal Ministry of Education and Research (BMBF) under Q-Exa (grant No. 13N16062) and QSolid (grant No. 13N16161). The authors also acknowledge the entire IQM Technology team for their support in the development of this work.



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toc




§ PRELIMINARIES

Here we detail and expand on some of the notation and technical details that go into showing the results claimed in the main text. Much of the structure for the circuits entering the protocol is inspired by <cit.>, so for this reason we stick to most of the original terminology and notation.



 §.§ Omega-distributed circuits

We will consider a quantum system made up of n≥2 qubits and a pair of single-qubit and two-qubit gate sets, G_1 and G_2, respectively, with corresponding probability distributions Ω_1 and Ω_2. We will then generate quantum circuits by sampling from 𝗀_1∼Ω_1 and 𝗀_2∼Ω_2, to generate parallel instructions on the n qubits as specified by a set L(G)={L_i=L_i^(1)L_i^(2)}, where each L_i^(1)∈L(G_1) and L_i^(2)∈L(G_2) are instructions made up of only the single and two-qubit gates 𝗀_1 and 𝗀_2, respectively. To stress when some quantities are random variables, we use sans font when relevant, as in 𝖫_i being a random layer sampled from Ω, which is such that Ω(𝖫_i)=Ω_1(𝖫_i^(1))Ω_2(𝖫_i^(2)).

Circuits generated this way, here for the particular case G=(G_1,G_2) and Ω=(Ω_1,Ω_2), are said to be Ω-distributed circuits, e.g., the circuit as stated in the main text as

    𝖢_m=𝖫_m𝖫_m-1⋯𝖫_1,

where each 𝖫_i∈L(Ω) sampled according to Ω, and the right-hand-side is read from right to left (i.e., 𝖢_m is an operator acting onto quantum states on the left), is a Ω-distributed circuit of depth m.

We point out that, in general, Ω-distributed circuits can be constructed with an arbitrary number of gate sets and their corresponding distributions. Moreover, while so far G_1,G_2 and Ω_1,Ω_2 are completely up to the user to specify, in practice the main requirements we will have on this choice are that, i) the resulting Ω-distributed circuits constitute an ϵ-approximate unitary 2-design, and ii) that in particular (G_1,Ω_1) is an exact unitary 2-design, e.g., the single-qubit Clifford group. As such conditions i) and ii) are considered a posteriori for a particular case of interest, we will have a general derivation and then these will be stated formally in point <ref> of  <ref> and in  <ref>, respectively.



 §.§ Ideal vs real implementations

While we do not stress this difference in the main text, here we will denote by U_L an ideal (noiseless) unitary operator associated to a representation of the layer L, and by U_L(ρ):=U_Lρ U_L^ the action of its corresponding map or superoperator U_L.

We write a composition of two maps A and B simply as BA to mean “apply map B after applying map A”, equivalent to the usual notations B∘A, or B(A(x)).

Noisy implementations of layers will generally be denoted by a map ϕ, e.g., the noisy application of a layer L is denoted by the cp map (which we also refer to as a quantum channel, or simply, a channel) ϕ(L), which can be defined (see e.g., IV of <cit.>) such that ϕ(L):=E_L U_L, with E_L being generally a cp map associated to L.



 §.§ The output states of Omega-distributed circuits

We will consider a set of Ω-distributed circuits of depth m, acting on a fiducial initial state |0⟩, which in turn we randomize with a layer 𝖵∈L(G_1). That is, we will have circuits of the form 𝖢_m𝖵|0⟩, which will always give a pure state output in the noiseless case. In the noisy case, however, we will have states of the form

    ϱ̃_m := _𝖫_i∼Ω
    𝖵∼Ω_1ϕ(𝖫_m)ϕ(𝖫_m-1)⋯ϕ(𝖫_1)ϕ(𝖵)(|0⟩⟨0|),

averaged independently over each of the Ω-distributed layers 𝖫_i, and uniformly over the initial single-qubit gate layer 𝖵∈L(G_1) with Ω_1. This noisy output can of course now be mixed, 2^-N≤(ϱ̃_m^2)≤1.

Our protocol focuses on efficiently estimating the purities of the states ϱ̃_m, to in turn estimate the average unitarity of noise in the circuits 𝖢_m. In the following, we first study the behavior of the average purity of ϱ̃_m in increasing circuit depth m, and then we see that we can efficiently estimate this quantity employing rms, allowing us to readily extract the average unitarity of noise in certain circumstances which we analyze.



§ PURITY OF AVERAGE OUTPUTS OF OMEGA-DISTRIBUTED CIRCUITS



 §.§ Purity of the initial randomization

Let us consider first the case of no gate layers but just the noisy initial single-qubit Clifford layer, m=0, then we have

    (ϱ̃_0^2)    = _𝖵∼Ω_1[ϕ(𝖵)(|0⟩⟨0|)ϕ(𝖵)(|0⟩⟨0|)^] 
       := _ψ∼Ω_1[E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|)E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|)^],

where we defined the noisy implementation of 𝖵 as ϕ(𝖵):=E_𝗌𝗉𝖺𝗆 U_𝖵 for some noise channel E_𝗌𝗉𝖺𝗆 explicitly standing for spam noise, and where we defined the initial randomized pure state as U_𝖵(|0⟩⟨0|):=|ψ⟩⟨ψ|.

Noticing that the noisy outputs E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|) are quantum states and thus Hermitian, and that we can define the self-adjoint map X^ of a quantum channel X by the property [AX(B)]:=[X^(A)B], we can write

    (ϱ̃_0^2)    = _ψ∼Ω_1⟨ψ|E_𝗌𝗉𝖺𝗆^E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|)|ψ⟩.

Equivalently, the self-adjoint map of a quantum channel can be defined in terms of its Kraus operators as X^(·):=∑K_μ^(·)K_μ, where {K_μ} the Kraus operators of X, so Eq. (<ref>) in a sense is already measuring how different E_𝗌𝗉𝖺𝗆 is from a unitary.

In general, the purity of a quantum state, ρ:=X(|φ⟩⟨φ|) can be written as (ρ^2):=⟨φ|X^X(|φ⟩⟨φ|)|φ⟩, which corresponds to the gate fidelity of X^X with respect to the identity on the state |φ⟩. Thus, we can write Eq. (<ref>) as a gate-fidelity of E_𝗌𝗉𝖺𝗆^E_𝗌𝗉𝖺𝗆 with respect to the identity, averaged over all possible initial states |ψ⟩,

    (ϱ̃_0^2)
           = _ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆^E_𝗌𝗉𝖺𝗆),

where here 

    _ψ(X):=⟨ψ|X(|ψ⟩⟨ψ|)|ψ⟩,

is the gate-fidelity of the map X with respect to the identity map on the state |ψ⟩.



 §.§ Relation to average unitarity and average trace-decrease

Ultimately, however, the average loss of purity in the average state ϱ̃_0 must be related to some amount of non-unitarity of the noise, i.e., it is noise that decreases purity. This can be quantified by the average unitarity, _|ψ⟩𝗎_ψ, defined for a cp map E as

    _ψ∼Ω_1𝗎_ψ(E):=(2^n2^n-1)_ψ∼Ω_1_ψ(E^' E^'),  where  E^'(·):=E(·-1/2^n),

which is so defined to account for the case of E being trace-decreasing. When E is tp we have the relation E^ 'E^'(·)=E^E(·-1/2^n), otherwise, however, for any quantum state ρ,

    E^ 'E^'(ρ) = E^E(ρ-12^n)-[E(ρ-12^n)]E^(12^n),

thus

    (2^n-12^n)_ψ∼Ω_1_ψ(E)    = _ψ∼Ω_1_ψ(E^E) 
           - _ψ∼Ω_1⟨ψ|E^E(1/2^n)|ψ⟩ - _ψ∼Ω_1[E^'(|ψ⟩⟨ψ|)]⟨ψ|E^(1/2^n)|ψ⟩
       = _ψ∼Ω_1_ψ(E^E) - 2^-n_ψ∼Ω_1{𝖲_ψ(E^E) + 𝖲_ψ(E)[𝖲_ψ(E) - [E(1/2^n)]]},

where

    𝖲_ψ(E):=[E(|ψ⟩⟨ψ|)],

is a trace-preservation measure of the map E for a given sample |ψ⟩.

We can then write the average purity of ϱ̃_0 as

    (ϱ̃_0^2)    = (2^n-12^n)_ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆) 
         + 2^-n_ψ∼Ω_1{𝖲_ψ(E_𝗌𝗉𝖺𝗆^E_𝗌𝗉𝖺𝗆) + 𝖲_ψ(E_𝗌𝗉𝖺𝗆)[𝖲_ψ(E_𝗌𝗉𝖺𝗆) - [E_𝗌𝗉𝖺𝗆(1/2^n)]]},

which we recall is defined as averaged over the states |ψ⟩⟨ψ|=U_𝖵(|0⟩⟨0|), which are distributed uniformly and independently on the n-qubits according to a distribution Ω_1 on single-qubit gates G_1.

When the noise channel E is tp, the rightmost summand in Eq. (<ref>) becomes a term equal to -2^-n, and in that case the average unitarity is just a re-scaling of the average gate-fidelity of the form

    _ψ∼Ω_1_ψ(E)  (E is TP)=  2^n _ψ∼Ω_1_ψ(E^E)-12^n-1.

and thus also the average purity in Eq. (<ref>) in this case would become

    (ϱ̃_0^2)    (E_𝗌𝗉𝖺𝗆 is TP)=(2^n-12^n)_ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆) + 12^n.




 §.§ The case of global depolarizing noise

A particular case of a quantum channel for modeling noise that is both easy mathematically as in interpretation is that of global depolarizing channel,

    E_p(·):=p (·)+(1-p)(·) 12^n,

where 0≤p≤1 is the probability for the input state to remain the same, and which can be directly defined in terms of the channel by the so-called polarization γ,

    γ(E) := 4^n(E)-14^n-1,

where here 

    (E):=⟨Ψ|(I⊗E)[|Ψ⟩⟨Ψ|]|Ψ⟩,    where  |Ψ⟩:=∑_i=1^2^n|ii⟩√(2^n),

is the so-called entanglement fidelity (equivalent to the purity of the so-called Choi state of E). Because maximally entangled states can be written in the Pauli basis as |Ψ⟩⟨Ψ|=4^-n∑_P∈𝖯𝖺𝗎𝗅𝗂_nP⊗P^T, this is equivalent to

    (E)=14^n∑_P∈𝖯𝖺𝗎𝗅𝗂_n12^n[PE(P)] = 14^n(E),

where we identified the matrix E, with elements E_ij=12^n[P_iE(P_j)], as the Pauli Transfer Matrix representation of E.

For a depolarizing channel, in particular,

    γ(E_p)=p.


Also, a composition of depolarizing channels with the same polarization is just another depolarizing channel as

    E_p⋯E_p_k times = E_p^k.


Given the above property, for a depolarizing channel, we also have

    _ψ(E_p)    = 2^n _ψ(E_p^2)-12^n-1
       = γ(E_p)^2 
       :=(E_p),

for any state |ψ⟩. To stress this independence of the state ψ, we drop the subindex in the last line, which simply means (E_p):=_φ(E_p) for an arbitrary pure state |φ⟩.

Notice that, E_p U_𝖢=U_𝖢 E_p, or more generally,

    E_p X=XE_p,

whenever X is a unital and tp quantum channel. Furthermore, for any unital and tp channel X, the unitarity _ψ(E_pX) only depends on _ψ(X^E_p^2X), which satisfies

    _ψ(X^E_p^2X)    = ⟨ψ|X^E_p^2X(|ψ⟩⟨ψ|)|ψ⟩
       = p^2_ψ(X^X) + (1-p^2)2^-n, 
       = (E_p)(_ψ(X^X)-2^-n)+2^-n,

and so

    _ψ(E_pX) = 2^n(E_p)(_ψ(X^X)-2^-n)2^n-1=(E_p)_ψ(X),

i.e. the unitarity this way for depolarizing channels factorizes.



 §.§ Average purity in the many-layer case

We now first consider adding a single layer of gates according to the distribution Ω, and evaluate the purity of ϱ̃_1,

    (ϱ̃_1^2)    = _𝖫_1∼Ω
    𝖵∼Ω_1[ϕ(𝖫_1)ϕ(𝖵)(|0⟩⟨0|)ϕ(𝖫_1)ϕ(𝖵)(|0⟩⟨0|)^] 
       = _𝖫_1∼Ω
    ψ∼Ω_1[E_1U_1E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|)E_1U_1E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|)] 
       = _𝖫_1∼Ω
    ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆^U_1^E_1^E_1U_1E_𝗌𝗉𝖺𝗆).


This can then be generalized directly to the case of m layers as

    (ϱ̃_m^2) = _𝖫_1∼Ω
    ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆^U_1^E_1^⋯U_m^E_m^E_mU_m⋯E_1U_1E_𝗌𝗉𝖺𝗆),

and thus by means of Eq. (<ref>), this is equivalent to

    (ϱ̃_m^2)
        = (2^n-12^n)_𝖫_i∼Ω
    ψ∼Ω_1_ψ(E_mU_m⋯E_1U_1E_𝗌𝗉𝖺𝗆) + 𝗍𝗋𝖺𝖼𝖾 𝖽𝖾𝖼𝗋𝖾𝖺𝗌𝗂𝗇𝗀 𝗍𝖾𝗋𝗆𝗌,

where the trace decreasing terms are terms in 𝖲 that can be read directly from Eq. (<ref>). One can express this way the most general form of the purity of the average state ϱ̃_m in circuit depth m.

However, we care, of course, for the situation in which the decay is exponential in m with the unitarity as the rate of decay.

    
  * We may consider first that the noise is approximately gate and time independent, E_i≈E, in which case at most we can get a double exponential in (E) and S(E), as in <cit.>. This approximation can be formalized by assuming that the average over time steps and gates is the leading contribution to the actual noise; see e.g., <cit.>.
    
  * If we consider noise that is only gate-independent but that is unital and tp, we have

    (ϱ̃_m^2)    = _𝖫_1∼Ω
    ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆^U_1^E_1^⋯U_m^E_m^E_mU_m⋯EU_1E_𝗌𝗉𝖺𝗆) 
       = _ψ∼Ω_1⟨ψ|_𝖫_i∼ΩE_𝗌𝗉𝖺𝗆^U_1^E_1^⋯U_m^E_m^E_mU_m⋯E_1U_1E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|)|ψ⟩,

and we can average recursively over each layer.
    
  *  This circuit averaging, however, becomes significant if the layer set at least forms an ϵ-approximate unitary 2-design [ A unitary t-design is a probability measure μ on the d-dimensional unitary group U(d) (or a subset thereof), such that T^(t)_μ = T_𝖧𝖺𝖺𝗋^(t), where T^(t)_μ(·):=∫_U(d)dμ(U)U^⊗t(·)(U^⊗t)^ is a t-twirl over the unitary group. There are several nonequivalent, albeit often related, ways of approximating unitary designs, here we employ that in <cit.>, where (1-ϵ)T_μ^(t)≼T^(t)_𝖧𝖺𝖺𝗋≼(1+ϵ)T^(t)_μ with A≼B here if and only if A-B is cp, which can be interpreted in a straightforward way as having a small multiplicative factor of difference when measuring a state acted on with T^(t)_μ vs with T^(t)_𝖧𝖺𝖺𝗋. Finally, for the case t=2, the 2-twirl on a cp map Φ can be stated in terms of its matrix representation Φ through T_μ^(2)(Φ), equivalent to the expression ∫_U(d)dμ(U)U^Φ(UρU^)U for any state ρ.], to the effect that _𝖫∼ΩU^_𝖫E^EU_𝖫≈Ξ̃_p for a global depolarizing channel Ξ̃_p^2 with polarization
    
    p^2    = ∑|[K_α^K_β]|^2-14^n-1
       =[E^E]-14^n-1
       = 4^n(E^E)-14^n-1
       = 2^n(E^E)-12^n-1,

    where in the first line {K_μ} are the Kraus operators of E <cit.>, or equivalently in the second line with E being a matrix representation of E [ It is possible to identify X=[     S(E)    X_𝗌𝖽𝗅;      X_𝗇 X_𝗎𝗇𝗂𝗍𝖺𝗅 ], where S(E), X_𝗌𝖽𝗅, X_𝗇 and X_𝗎𝗇𝗂𝗍𝖺𝗅 are trace-decreasing, state-dependent leakage, non-unital and unital components (of dimensions 1, 1×(4^n-1), (4^n-1)×1 and square (4^n-1)), respectively. This leads to the expression in <cit.> for the average unitarity.], or in terms of the entanglement fidelity through Eq. (<ref>) in the third line, or finally through the average gate-fidelity <cit.>, as we write in the main text. Thus, if we further consider Ω-distributed circuits that form an approximate unitary 2-design, we may write
    
    (ϱ̃_m^2)    ≈_ψ∼Ω_1⟨ψ|E_𝗌𝗉𝖺𝗆^Ξ̃_p_m^2⋯Ξ̃_p_1^2E_𝗌𝗉𝖺𝗆(|ψ⟩⟨ψ|)|ψ⟩,

where the approximation “≈” here neglects the small ϵ multiplicative corrections; due to the property in Eq. (<ref>), and where we intentionally defined the polarizations to be squares, given that p^2=(Ξ̃_p_i) [ We notice that this does not mean that Ξ̃_p corresponds to (approximately) the average of a single E (which only occurs if E itself is already depolarizing).]. For approximately time-stationary noise, to the effect that all polarizations are equal, p_1=⋯=p_m=p, this is equivalent to the exponential decay

    (ϱ̃_m^2)    ≈ A(Ξ̃_p)^m+12^n,

where A=2^-n(2^n-1)_ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆) is a spam constant. Of course, given that (Ξ̃_p)=p^2, this unitarity can also be expressed by (E), as done in Eq. (<ref>) in the main text.




§ OPERATIONAL ESTIMATION OF THE PURITY

Now we know how the purity of the noisy average state ϱ̃_m in Eq. (<ref>) behaves in terms of the average unitarity of noise with respect to circuit depth m. The task is then to estimate this quantity via rms.



 §.§ Purity estimation through randomized measurements

rms found some of their first applications in estimating the purity of the quantum state of a subsystem <cit.>. Specifically, here we will focus on the results of <cit.>, whereby it is shown that for a reduced quantum state ρ_A=_B(ρ_AB), from some composite closed quantum system AB, where here we assume A to be a n-qubit quantum system, its purity can be expressed as

    (ρ_𝖠^2) = 2^n∑_𝐬,𝐬^' (-2)^-h(𝐬,𝐬^')_𝖴∼𝖧𝖺𝖺𝗋[𝖯_𝖴(𝐬)𝖯_𝖴(𝐬^')],

where 𝐬 and 𝐬^' are n-bit strings, h(𝐬,𝐬^') is the Hamming distance (number of distinct bits) between them, _𝖴∼𝖧𝖺𝖺𝗋 denotes uniform averaging with the Haar measure over the local unitaries 𝖴=_i=1^n𝖴_i, [ That is, E_𝖧𝖺𝖺𝗋 here refers to a uniform and independent average over all the local unitaries U_i with the Haar measure, i.e., E_𝖧𝖺𝖺𝗋∼E_U_1E_U_2⋯E_U_N] and the

    𝖯_𝖴(𝐬):=⟨𝐬| U_𝖴(ρ_𝖠)|𝐬⟩,

are noiseless probabilities of observing the n-bit string 𝐬 upon applying the unitary 𝖴 on the reduced state ρ_𝖠.

The main advantage here is that the quantum component of estimating purity this way is just the probabilities in Eq. (<ref>), which simply means applying a product of random unitaries and measuring each on a user-specified basis. The rest, i.e., fully estimating Eq. (<ref>) is purely classical post-processing of said probabilities. For n qubits, the number of distinct terms in the sum over n-bit strings in Eq. (<ref>) is 2^n-1(2^n-1), since the Hamming distance is symmetric, h(𝐬,𝐬^') = h(𝐬^', 𝐬); however, this sum is only a function of the probabilities 𝖯_𝖴, so it could, for example, be computed in parallel.



 §.§ Purity of many layer Omega-distributed circuits through randomized measurements

Let us again begin by considering the case m=1, of a single random layer, so that the equivalent of the probability in Eq. (<ref>), by measuring the state ϱ̃_1 with a rm using a random unitary 𝖶∈L(G_1) over Ω_1, reads as

    𝖯_1,𝖶(𝐬)    = [ϕ(𝖶)(|𝐬⟩⟨𝐬|) ϱ̃_1] 
       = _𝖫_1∼Ω
    𝖵∼Ω_1[ϕ(𝖶)(|𝐬⟩⟨𝐬|) ϕ(𝖫_1)ϕ(𝖵)(|0⟩⟨0|)] 
       =_𝖫_1∼Ω
    ψ∼Ω_1[E_𝗌𝗉𝖺𝗆_𝗐^'U^'_𝖶(|𝐬⟩⟨𝐬|) E_1 U_1 E_𝗌𝗉𝖺𝗆_𝗏(|ψ⟩⟨ψ|)] 
       =_𝖫_1∼Ω
    ψ∼Ω_1⟨𝐬| U_𝖶 E_𝗌𝗉𝖺𝗆_𝗐 E_1 U_1 E_𝗌𝗉𝖺𝗆_𝗏(|ψ⟩⟨ψ|)|𝐬⟩,

where we again defined U_𝖵(|0⟩⟨0|)=|ψ⟩⟨ψ| as a random (according to a uniform U_𝖵∼Ω_1) pure state, and identified each E_𝗌𝗉𝖺𝗆 as the noise contribution to spam; in the penultimate line (purely for notational reasons) we defined ϕ(𝖶):=E_𝗌𝗉𝖺𝗆_𝗐^'U^'_𝖶 and in the last line, U_𝖶=U_𝖶^' and E_𝗌𝗉𝖺𝗆_𝗐=E_𝗌𝗉𝖺𝗆_𝗐^'.

Putting this together with the probability of getting another n-bit string 𝐬^' with the same layer sequence,

    𝖯_1,𝖢(𝐬)𝖯_1,𝖢(𝐬^')
           = _𝖫_1∼Ω
    ψ∼Ω_1⟨𝐬|U_𝖶 E_𝗌𝗉𝖺𝗆_𝗐 E_1 U_1 E_𝗌𝗉𝖺𝗆_𝗏(|ψ⟩⟨ψ|)|𝐬⟩⟨𝐬^'| U_𝖶 E_𝗌𝗉𝖺𝗆_𝗐 E_1 U_1 E_𝗌𝗉𝖺𝗆_𝗏(|ψ⟩⟨ψ|)|𝐬^'⟩
       = _𝖫_1∼Ω
    ψ∼Ω_1⟨𝐬| U_𝖶 E_𝗌𝗉𝖺𝗆_𝗐 E_1 U_1 E_𝗌𝗉𝖺𝗆_𝗏[|ψ⟩⟨ψ|E_𝗌𝗉𝖺𝗆_𝗏^ U_1^E_1^ E_𝗌𝗉𝖺𝗆_𝗐^ U_𝖶^(|𝐬^'⟩⟨𝐬^'|)|ψ⟩⟨ψ|]|𝐬⟩
       = _𝖫_1∼Ω
    ψ∼Ω_1⟨𝐬| U_𝖶 E_𝗌𝗉𝖺𝗆_𝗐 E_1 U_1 E_𝗌𝗉𝖺𝗆_𝗏P_ψE_𝗌𝗉𝖺𝗆_𝗏^ U_1^E_1^ E_𝗌𝗉𝖺𝗆_𝗐^ U_𝖶^(|𝐬^'⟩⟨𝐬^'|)|𝐬⟩,

where in the last line we defined P_ψ(·):=|ψ⟩⟨ψ|·|ψ⟩⟨ψ|.

Now we can exploit the identity derived in <cit.>,

    (X)   = ∑_x(-2)^-h(x,y)⟨x|_𝖴_i∼𝖧𝖺𝖺𝗋{U_𝖴^ X U_𝖴} [|y⟩⟨y]|x⟩,

where here  is entanglement fidelity, defined in Eq. (<ref>), and U_𝖴 is the noiseless unitary map corresponding to the product 𝖴=_i=1^n𝖴_i with local, single-qubit random unitaries 𝖴_i. Since this expression requires two copies of U_𝖴, where only the individual 𝖴_i should be Haar random, it suffices for these (as opposed to the global unitary) to belong to a unitary 2-design, e.g., the single-qubit Clifford group.

We emphasize then, that we will require the rms to be randomized by a layer made up of uniformly random single-qubit Clifford gates; while in the main text we fix this to correspond to G_1 and Ω_1, all other layers in principle can be chosen arbitrarily (with the reasoning of having some better choices than others detailed in the previous sections).

We can now apply Eq. (<ref>) on Eq. (<ref>), so that

    (ϱ̃_1^2)    ≃ 4^n _𝖫_1∼Ω
    ψ∼Ω_1(E_𝗌𝗉𝖺𝗆_𝗐E_1U_1E_𝗌𝗉𝖺𝗆_𝗏P_ψE_𝗌𝗉𝖺𝗆_𝗏^ U_1^E_1^E_𝗌𝗉𝖺𝗆_𝗐^) 
       = _𝖫_1∼Ω
    ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆_𝗏^ U_1^E_1^E_𝗌𝗉𝖺𝗆_𝗐^E_𝗌𝗉𝖺𝗆_𝗐E_1 U_1E_𝗌𝗉𝖺𝗆_𝗏),

where the second line follows directly from the definition of entanglement fidelity. This generalizes to any circuit depth m, to

    (ϱ̃_m^2) = _𝖫_i∼Ω
    ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆_𝗏^ U_1^E_1^⋯U_m^E_m^E_𝗌𝗉𝖺𝗆_𝗐^E_𝗌𝗉𝖺𝗆_𝗐E_m U_m⋯E_1 U_1E_𝗌𝗉𝖺𝗆_𝗏),

which coincides with Eq. (<ref>) up to E_𝗌𝗉𝖺𝗆_𝗐.

Now similar to point <ref> in  <ref> above, we mainly care about an exponential decay, which occurs for Ω-distributed circuits being an approximate unitary 2-design and noise being approximately gate+time independent, unital and tp. Now we will have

    (ϱ̃_m^2) ≈ A(Ξ̃_p)^m + 12^n,

for A=2^-n(2^n-1)(Ξ̃_q_𝗌𝗉𝖺𝗆_w)E_ψ∼Ω_1_ψ(E_𝗌𝗉𝖺𝗆_𝗏), where Ξ̃_q_𝗌𝗉𝖺𝗆_𝗐 is the contribution from the layer average of E_𝗌𝗉𝖺𝗆_𝗐; of course both unitarity terms in A cannot be distinguished and can simply be interpreted as an average unitarity of spam.

The result in Eq. (<ref>) corresponds to that with an exact average over layers and initial states; this average can be approximated numerically with Eq. (<ref>) given a number N_𝖶 of measurements, 𝖢_m of Ω-distributed circuits, and N_𝖵 of initial randomization samples, respectively.














§ OPERATIONAL ESTIMATION OF FIDELITY THROUGH RANDOMIZED MEASUREMENTS

Since estimating the average unitarity is really useful only when informed by the average fidelity, we now consider estimating the average layer fidelity of the same Ω-distributed circuits, once given a set of randomized measurement probabilities 𝖯_𝖴.

Naturally, if the noiseless output is a state |φ⟩,  Eq. (<ref>) can be modified to account for its fidelity with respect to a mixed state ρ as

    ⟨φ|ρ|φ⟩ = 2^n∑_𝐬,𝐬_𝗂𝖽𝖾𝖺𝗅 (-2)^-h(𝐬,𝐬_𝗂𝖽𝖾𝖺𝗅)_𝖴∼𝖧𝖺𝖺𝗋𝖯_𝖴(𝐬)𝖯^(𝗂𝖽𝖾𝖺𝗅)_𝖴(𝐬_𝗂𝖽𝖾𝖺𝗅),

where 𝖯^(𝗂𝖽𝖾𝖺𝗅)_𝖴 is the probability of observing a n-bit string 𝐬_𝗂𝖽𝖾𝖺𝗅 with a randomized measurement defined by 𝖴 on |φ⟩⟨φ|, that is

    𝖯^(𝗂𝖽𝖾𝖺𝗅)_𝖴(𝐬_𝗂𝖽𝖾𝖺𝗅) = |⟨𝐬_𝗂𝖽𝖾𝖺𝗅|𝖴|φ⟩|^2.


Notice that this assumes that we can efficiently compute the n-bit strings, and thus the probabilities, corresponding to the noiseless outputs. This generally implies a limitation in the gate sets that can be employed to only Clifford gates, however, for mid-scale estimations the ideal probabilities can still be computed without a major overhead when the gate set contains non-Clifford gates to be sampled with a relatively low probability.

In our case, we want to estimate the quantity ⟨ψ_m|ϱ̃_m|ψ_m⟩, where

    |ψ_m⟩⟨ψ_m| := 𝖢_m^(𝗂𝖽𝖾𝖺𝗅)(|0⟩⟨0|)    = U_m⋯U_1U_𝖵(|0⟩⟨0|) 
       = U_m⋯U_1(|ψ⟩⟨ψ|).


Thus, consider first the case m=1; we have

    𝖯_1,𝖶(𝐬)𝖯^(𝗂𝖽𝖾𝖺𝗅)_1,𝖶(𝐬_𝗂𝖽𝖾𝖺𝗅)    = _𝖫_1∼Ω
    ψ∼Ω_1⟨𝐬| U_𝖶 E_𝗌𝗉𝖺𝗆_𝗐 E_1 U_1 E_𝗌𝗉𝖺𝗆_𝗏P_ψ U_1^ U_𝖶^(|𝐬_𝗂𝖽𝖾𝖺𝗅⟩⟨𝐬_𝗂𝖽𝖾𝖺𝗅|)|𝐬⟩,

which follows as in Eq. (<ref>), where in the last line we defined P_ψ(·):=|ψ⟩⟨ψ|·|ψ⟩⟨ψ|. Similarly, we can follow all steps from Eq. (<ref>) to Eq. (<ref>) in an analogous way to obtain

    ⟨ψ_m|ϱ̃_m|ψ_m⟩ = _𝖫_i∼Ω
    ψ∼Ω_1_ψ( U_1^⋯U_m^E_𝗌𝗉𝖺𝗆_𝗐E_m U_m⋯E_1 U_1E_𝗌𝗉𝖺𝗆_𝗏),

which is simply an rb-like fidelity decay. Similarly here, if the Ω-distributed circuits approximate a unitary 2-design, to the effect that _𝖫∼ΩU^_𝖫E_iU_𝖫≈Ξ̃_p_i, i.e., the average noise is approximately a global depolarizing channel with p_i=(2^n(E)-1)/(2^n-1), we end up with

    ⟨ψ_m|ϱ̃_m|ψ_m⟩   ≈_ψ∼Ω_1_ψ(Ξ̃_p_m^'Ξ̃_p_m-1⋯Ξ̃_p_1E_𝗌𝗉𝖺𝗆_𝗏) 
       = _ψ∼Ω_1_ψ(Ξ̃_p_𝗌𝗉𝖺𝗆_𝗐Ξ̃_p_mp_m-1⋯p_1E_𝗌𝗉𝖺𝗆_𝗏),

 which is a rb-like decay of the state fidelity in the circuit depth m, where we defined p_m^'=p_mp_𝗌𝗉𝖺𝗆_𝗐 for some p_𝗌𝗉𝖺𝗆_𝗐. Further, for time-independent noise, such that p_1=p_2=…=p_m=p, this is an exponential decay in m,

    ⟨ψ_m|ϱ̃_m|ψ_m⟩   ≈ Ap^m + B,

where A=p_𝗌𝗉𝖺𝗆_𝗐[_ψ(E_𝗌𝗉𝖺𝗆_𝗏)-2^-n], B=2^-n, and

    p = 2^n(E)-12^n-1,

which can similarly be written in terms of entanglement fidelity of E, or in terms of its matrix representation, similar to the usual rb constants for unital spam.

This means we can generate a set of random Ω-distributed circuits, attach some set of randomized measurements, and use the probabilities to simultaneously estimate average layer fidelity and unitarity.



§ SCRAMBLING AND UNITARY 2-DESIGNS


Here we refer to both assumptions, <ref>, <ref>, as written in the main text in  <ref>, for an exponential decay of the average sequence purity in circuit depth.

While assumption <ref> is beyond the choices that the user executing the protocol <ref> can make, assumption <ref> is important to make the protocol <ref> adhere to the exponential decay of main Result <ref>, but it also sets a practical limit in scalability to mid-scale systems, which we discuss below.

There is, furthermore, a fundamental limit to scalability imposed by the sample complexity of estimating purity via rms, i.e., the number of experimental samples, N_𝗆𝖾𝖺𝗌 in Eq. (<ref>), needed to estimate purity within a given error. This is discussed in the main text in  <ref>, and it compounds with the sample complexity of the Ω-distributed circuits, related to assumption <ref>.

The need for unitary 2-designs in rb-based techniques stems from the fact that, by definition, these inherit the property of fully random (i.e. Haar distributed) unitaries whereby their second moment is described by a depolarizing channel, which requires a single parameter to be fully specified. This implies that figures of merit, such as average gate fidelity or average gate unitarity can be encapsulated into such polarization parameter (once assumption <ref> is satisfied) given circuits that generate a unitary 2-design.

Loosely speaking, a unitary 2-design is a probability distribution μ on the unitary group, or a subset thereof, satisfying _V∼μV^XV = _U∼𝖧𝖺𝖺𝗋U^XU for any quantum channel X. The action _V∼μV^(·)V is referred to as a 2-twirl with μ. That is, the 2-twirl with μ reproduces the second moment of the whole unitary group with the Haar measure. That an ensemble of unitaries constitutes a 2-design is particularly useful because it is known that doing a 2-twirl with the Haar measure on a cp map, reduces it to a depolarizing channel <cit.>, i.e., also for a 2-design _V∼μV^XV(·) = p (·)+(1-p)1/2^n, where here p=(2^n(X)-1)/(2^n-1), where (X) is the average gate fidelity of X. Both Eq. (<ref>) and Eq. (<ref>) are obtained using this relation.

The notion of a unitary 2-design can further be relaxed by having equality up to a small ϵ according to a given metric, to the effect that the 2-twirl is also close to a depolarizing channel. More precisely, denoting a 2-twirl by Δ_μ(·):=_V∼μV^(·)V we say that V is an ϵ-approximate unitary 2-design, if

    (1-ϵ)Δ_𝖧𝖺𝖺𝗋≼Δ_μ≼(1+ϵ)Δ_𝖧𝖺𝖺𝗋,

where here ≼ is semidefinite ordering in the sense that X≼Y if and only if X-Y is a cp map. This definition was put forth (for the more general setting of t-designs) in <cit.> and has the particularly simple interpretation of having a small multiplicative factor 1±ϵ of difference when measuring a state acted on with a channel twirled Δ_μ as opposed to with Δ_𝖧𝖺𝖺𝗋 [ This definition is further connected in <cit.> to the usual diamond norm definition as implying that if V is an ϵ-approximate unitary 2-design, also Δ_μ-Δ_𝖧𝖺𝖺𝗋_♢≤2ϵ, where X_♢:=sup_ρ(I_d⊗X)ρ_1, where X_1=√(XX^) denotes trace norm and the supremum is taken over all dimensions d≥1 of the identity and corresponding density matrices ρ.]. Thus, the approximation sign in both Eq. (<ref>) and Eq. (<ref>) refers to these small multiplicative factors in the 2-design approximation, exponential in m.

A well-known example of an exact 2-design is the n-qubit Clifford group <cit.>: it is also simultaneously a strong reason
why standard [ As can be seen in <cit.> and <cit.>, there is a plethora of techniques under the term rb; by standard we mean single or two-qubit RB with corresponding Clifford gate sets, estimating the group's average gate fidelity by fitting survival probabilities to an exponential decay in sequence length.] Clifford rb works so well, and the culprit (in practice) of why it does not scale in n. Techniques akin to standard Clifford rb, such as drb, mrb and birb resolve such scalability issues by, among other things, retaining some of the randomness through Ω-distributed circuits. Rather than requiring an approximate unitary design property, e.g., mrb <cit.> and birb <cit.> require a scrambling property in the sense that Pauli errors get quickly spread among other qubits before other Pauli errors occur. This can be stated in terms of an entanglement fidelity for non-identity Paulis P, P^', with action P(·)=P(·)P and P'(·)=P^'(·)P^', such that for an expected infidelity of the layers α, there is k≪1/α and a δ≪1 with

    _Ω[P C_Ω P^' C_Ω^-1] ≤δ + 14^n,

where here C_Ω=U(𝖫_k⋯𝖫_1), with U the unitary map of the sequence of noiseless layers 𝖫_k⋯𝖫_1. Similarly, drb uses generators of unitary 2-designs as the benchmarking gate set and then requires that these constitute a sequence-asymptotic unitary 2-design <cit.>.

While the scrambling condition in Eq. (<ref>) is stated for Pauli noise channels, the choice of (G_1,Ω_1) being the uniformly distributed single-qubit Clifford group has the effect of projecting any cp map (modeling Markovian noise) to a Pauli channel (i.e., a tensor product of depolarizing channels) since each Clifford group constitutes a unitary 2-design on the respective qubit.

The reason approximating a unitary 2-design is a stronger condition than scrambling, as defined by Eq. (<ref>), is because it would require δ shrinking as O(4^-n) [ This can be seen e.g., by assuming the U(𝖫_k⋯𝖫_1) generates a unitary 2-design and using Eq. (<ref>)]. Nevertheless, while we directly assume that our Ω-distributed circuits approximate a unitary 2-design, for a mid-scale of tenths of qubits the scrambling condition suffices, and indeed it holds in a similar way than it does in mrb or birb [ A difference of our technique with mrb is that we do not employ a mirror structure, while one with birb is that we do not employ initial and final stabilizers.], as exemplified in  <ref>. Increasing the qubit count to estimate average unitarity would require establishing a scrambling condition, e.g., as for birb by employing stabilizer states and measurements, or otherwise relaxing the approximate design condition.



§ NUMERICAL ADDENDUM



 §.§ The simulator backend

As explained in the main text in  <ref>, we performed experiments on , a 5-qubit superconducting system with a central qubit connected to the rest, and simulations with a backend of a total of 20 qubits in a grid topology, connected as depicted in the graph of Fig. <ref>. In the simulation, we considered a subset of these qubits, namely qubits ranging from qubit 3 to qubit 12, in pairs as depicted in Fig. <ref>. All qubits were associated with a noise model on readout, gate times, single- and two-qubit depolarizing noise, and decoherence, T_1 and T_2 times. 



 

 §.§ Simulation on 10 qubits


We now employ a simulator backend that considers 20 possible qubits with a grid topology, as displayed fully in Fig. <ref> of Appendix <ref>. All operations (except barriers) are modeled as noisy, in this particular example with a set of parameters considering a range of relaxation times T_1, dephasing times T_2, gate duration parameters, depolarizing parameters on both single- and two-qubit gates, and readout errors (bit-flip probabilities).

Similarly in this case, according to Protocol <ref>, we extracted decay rates of the average state purity and fidelity in increasing depths, and thus average layer unitarity and fidelity for Ω-distributed circuits with up to 10 qubits. In Fig. <ref>, we show the corresponding purity decays in increasing depth. We display only even qubit numbers n for easiness of visualization, where each n corresponds to choices of qubits and couplings as shown in the inset. In all cases, we employed unbiased estimators and K=2 momsmomsmoms estimators.



 In Fig. <ref> we show all average layer fidelities and unitarities in increasing number of qubits, together with the Pauli noise unitarity intervals as predicted by the bound in Ineq. (<ref>). In the corresponding inset, we show the average state fidelity decays in increasing depth for each number of qubits n, and thus the corresponding average gate fidelities; similar to the case of , these were computed using the measurements within the same experiment and using the estimator in Eq. (<ref>) using the measurements of the noiseless circuits. The noiseless probabilities were estimated with the same number of N_𝗆𝖾𝖺𝗌=2^10 shots per measurement.

Generally, while variances in the decays are relatively high, the averages show only a modest deviation up to 9 qubits. The noise model is dominated by stochastic errors, so it is expected that most points should fall within the Pauli unitarity bound; the cases of n=6,8 are indeed outliers, which in experiment would need to be investigated further, in our case they most likely can be explained by both the qubit arrangement and the amplitude damping T_1 parameters involved. It should be noted that for simulation in a common laptop, going beyond 10 qubits becomes quite computationally demanding.



 §.§ Comparison of layer fidelity estimations with MRB

The Ω-distributed circuits we consider are a primitive component in mrb <cit.>, thus the average fidelity estimations from both our technique and mrb should agree. There are two main differences between both procedures: one is that we do not construct a “mirror” circuit, i.e., we do not append the circuit made up of the inverses of the original Ω-distributed circuit, and the second is that we do not interleave layers of random Pauli gates. Our technique is able to estimate average unitarity too, although this comes at a sampling complexity cost that requires exponentially many more samples to increase accuracy.



  §.§.§ IQM Spark (TM)

Tthe estimated fidelities from randomized measurements should be consistent with those estimated by mrb <cit.>. Indeed, a primitive of mrb is Ω-distributed circuits; a difference with our technique is that we do not implement a mirror (i.e. the sequence of inverses), nor interleave random Pauli operators. Nevertheless, both techniques can estimate the same quantity (mrb through polarizations instead of state fidelities), rms simply enable to access the unitarity. In Fig. <ref> we show the fidelity decays obtained experimentally with mrb about a month ago on , with the same Ω-distributed circuits considered in  <ref>, namely made up of single-qubit Clifford gates uniformly sampled and 𝖢𝖹 gates with a two-qubit gate density of 1/2 according to the edge-grab sampler.



Clearly, while the estimated average layer fidelities agree, the distributions have a smaller variance than those in Fig. <ref>, and were obtained with less total circuit samples. It can be conceivable then, that alternatively, average fidelities may be computed by constructing mirror circuits and following the mrb protocol <cit.>, at the expense of performing separate measurements of these circuits.



§ UNITARITY AND RANDOMIZED COMPILING

To begin with, let us label the n-qubit Pauli group P_n, i.e., the group made up of all n-fold products of Pauli operators (1,X,Y,Z), as in <cit.>: let a be a 2n-bit string, a=a_1a_2…a_2n, and define

    P_a := i^a^TΥa∏_j=1^nX_j^a_2j-1Z_j^a_2j,

where X_j and Z_j are single-qubit Pauli operators acting on the jth qubit and Υ:=_j=1^n([ 0 1; 0 0 ]) such that P_a is ensured to be Hermitian. The Pauli group P_n is then made up of all such P_a together with their products and their overall phases {±1,±i}; e.g., P_1={±1,±i1,±X,…,±iZ}. In particular, denoting any other 2n-bit strings with bold lowercase letters, we will make use of the property

    P_aP_b = (-1)^⟨a,b ⟩P_bP_a,

where

    ⟨a,b ⟩   := a^T(Υ+Υ^T)b2
       = (a_1b_2 + a_2b_1 + a_3b_4 + a_4b_3 + …+a_2nb_2n-1+a_2n-1b_2n)2,

which in particular is such that

    14^n∑_a(-1)^⟨a,b+c ⟩ = δ_bc .


Let us then write a generic n-qubit noise channel (i.e., some cp map) in the so-called χ-representation, E(·)=∑_a,bχ_abP_a(·)P_b, where χ is a positive Hermitian matrix. We can also relate this representation to the so-called ptm representation, which is a 4^n-square matrix with real entries given by

    E_ab   := 12^n[P_aE(P_b)] 
       = 12^n∑χ_cd[P_aP_cP_bP_d],

which is not quite informative by itself; however, a particular case where this representation is useful is for the case of a Pauli channel, which has for χ a diagonal matrix, implying that its ptm is diagonal too. A Pauli channel can be enforced on any channel by twirling it with the Pauli group, i.e.,

    E ↦ E^(·) := 14^n∑_P∈P_nPE(P·P)P,

which gives

    E^_ab   = 12^3n∑_c,d,qχ_cd[P_aP_qP_cP_qP_bP_qP_dP_q] 
       = 12^3n∑_c,d,q   (-1)^⟨q,c+d⟩ χ_cd [P_aP_cP_bP_d] 
       = 12^n∑_c,dχ_cdδ_cd[P_aP_cP_bP_d] 
       = 12^n∑_dχ_dd[P_aP_dP_bP_d] 
       = δ_ab∑_d  (-1)^⟨a,d⟩χ_dd := δ_ab∑_d  (-1)^⟨a,d⟩α_d,

where we made use of both Eq. (<ref>) and Eq. (<ref>), so that indeed E^, the ptm of the twirled map E^, is manifestly diagonal with eigenvalues λ_a:=∑_b  (-1)^⟨a,b⟩α_b, where we write the diagonal elements with a single index as α_a:=χ_aa.

The diagonal elements α are the so-called Pauli error probability rates, conversely related to the ptm eigenvalues as α_a=2^-n∑_b(-1)^⟨a,b⟩λ_b, and in terms of these, the trace non-increasing condition translates to ∑_aα_a≤1, saturating for E being tp.

If follows that the average gate fidelity of a Pauli channel is given by

    (E^)    = [E^]+2^n2^n(2^n+1)
       = ∑_aλ_a+2^n2^n(2^n+1)
       = 2^nα_0+12^n+1
       = (E) 
       E is TP= 1 - (2^n2^n+1)∑_i≠0α_i,

where in the penultimate line we highlighted that it is the same as for the raw (not-twirled) channel E, and where used Eq. (<ref>), with 0 being the 2n-bit string with all bits being zero, and in the last line the case ∑_aα_a=1 for E being tp; such expression makes it manifest that average gate fidelity captures only the information of the probability of any Pauli error happening [ This does not mean that average gate fidelity is insensible to coherent errors, but rather that it is only partially so through their contribution to the diagonal elements in the ptm.].

Conversely, the unitarity of the Pauli channel is given by

    (E^)    = [E^ E^]-14^n-1
       = ∑_aλ_a^2-14^n-1
       = 4^n∑_aα_a^2-14^n-1,

which is minimal in the sense that for a non-identity it corresponds to a purely stochastic noise channel, as it only involves the diagonal χ-matrix terms, the Pauli error rates, involved in the average gate-fidelity. Clearly, when the distribution of errors is uniform, i.e., all α_i=2^-n so that the channel is maximally depolarizing, the unitarity vanishes. As opposed to average gate fidelity, the average unitarity of the raw channel E would capture all elements of its ptm. We can further lower-bound this as follows,

    (E^)    = 4^n∑_aα_a^2-14^n-1
       ≤4^n[α_0^2+(∑_b≠0α_b)^2]-14^n-1
       ≤4^n[α_0^2+(1-α_0)^2]-14^n-1
       = (2^n(E)-12^n-1)^2 + (4^n-2)(1-(E)2^n-1)^2,

so together with the lower bound f(E)^2 of <cit.>, we can express the unitarity for the Pauli-twirled channel E^ as

    f(E)^2 ≤(E^) ≤f(E)^2 + 2^2n-2(2^n-1)^2 r(E)^2,

where here f(E)=(2^n(E)-1)/(2^n-1) is the noise-strength parameter in Eq. (<ref>) in the main text, and r(E):=1-(E) is the average gate infidelity of E.

The lower bound in Ineq. (<ref>) saturates for depolarizing noise, i.e., when all non-identity Pauli error rates are the same, while the upper bound overestimates the unitarity for Pauli noise through cross-products of non-identity Pauli error rates. Thus, the upper bound serves as a proxy for whether noise has coherent components, if it is above it, or whether it is potentially Pauli, if it is under it.

Of course, in practice, we cannot directly twirl noise quantum channels but we need a way of achieving it by manipulating the noisy gates in question. Whenever the ideal (or target) gates are Clifford, a Pauli twirl can be enforced by a so-called G-twisted twirl <cit.>, where the outer Pauli operators in Eq. (<ref>) are acted on (or twisted) with the ideal Cliffords. This concept of a twisted-twirl, together with employing random samples instead of all the 4^n possible Pauli operators, are at the core of so-called rc, where we would then effectively have

    E^_N(·) := 1N∑_P∼P_n^N samplesPE(P·P)P,

where here P∼𝖯𝖺𝗎𝗅𝗂 means P are Pauli samples drawn uniformly at random with no repetition from P_n, and the sum runs over N≤4^n such samples. In practice, this randomized sum would be computed by measuring and then averaging many circuits where the noisy gate is twisted-twirled; considering that the noisy gate may be transpiled into single and two-qubit gates, the way this is usually done is by precisely compiling the single-qubit gates with the Paulis in each random sample, with the overall gate being logically exactly the same. Since also in practice the Pauli operators will be noisy, care has to be taken not to increase the overall number of gates that are executed. In general if the Paulis themselves have coherent noise, perfect diagonalization would be achieved strictly for N→∞ by sampling with repetition.

The corresponding ptm of E^_N(·) has components

    (E^_N)_ab   = 1N2^n∑_q^N samples∑_c,dχ_cd[P_aP_qP_cP_qP_bP_qP_dP_q] 
       = 12^n∑_c,d(1N∑_q^N samples(-1)^⟨q,c+d⟩) χ_cd [P_aP_cP_bP_d],

where now we can only approximate the property in Eq. (<ref>) with the N random samples of bit strings q: in the infinite sample limit, N→∞, ,or if exactly each distinct bit string is sampled once, the term within parenthesis goes to δ_cd. Since ⟨q,2c⟩=0 for any pair of bit strings q and c, the sum over q when c=d equals 1 for any number of samples; this just reflects the fact that twirling (or rc) leaves the diagonal of the ptm invariant. Thus,  we can write

    (E^_N)_ab   = E^_ab + ∑_c≠d(1N∑_q^N samples(-1)^⟨q,c+d⟩ χ_cd [P_aP_cP_bP_d]2^n) 
       := E^_ab + 1N∑_q^N samplesλ̃_ab^(q),

where here E^ as before, is the perfectly Pauli-twirled channel in Eq. (<ref>), and where we defined 

    λ̃_ab^(q):=∑_c≠d(-1)^⟨q,c+d⟩ χ_cd [P_aP_cP_bP_d]2^n,

which correspond to the off-diagonal elements of the ptm upon the twirl action of an element P_q. The effect of N=1, a single randomization, will simply be to either change a sign of an off-diagonal element or leave it as it is. For an even number N of samples, the off-diagonals will either vanish or be suppressed in magnitude as integer multiples of 2/N (with ratio less than one). On the other hand, for odd N, all off-diagonals vanish in magnitude as integer multiples of 1/N.

Finally, then

    (E^_N)    = (E^) + 1N^2∑_q,q^'^N samplesλ̃_ab^(q)λ̃_ba^(q^')4^n-1,

where similarly now, the off-diagonal contributions will vanish as (small) multiples of 1/N^2.

