# Quantum State Compression Shadow

Chen Ding  Henan Key Laboratory of Quantum Information and Cryptography,
Zhengzhou, Henan 450000, China  Xiao-Yue Xu  Henan Key Laboratory of Quantum
Information and Cryptography, Zhengzhou, Henan 450000, China  Shuo Zhang
Henan Key Laboratory of Quantum Information and Cryptography, Zhengzhou, Henan
450000, China  Wan-Su Bao<EMAIL_ADDRESS>Henan Key Laboratory of Quantum
Information and Cryptography, Zhengzhou, Henan 450000, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei 230088, China
He-Liang Huang<EMAIL_ADDRESS>Henan Key Laboratory of Quantum
Information and Cryptography, Zhengzhou, Henan 450000, China  Hefei National
Laboratory, University of Science and Technology of China, Hefei 230088, China

###### Abstract

Quantum state readout serves as the cornerstone of quantum information
processing, exerting profound influence on quantum communication, computation,
and metrology. In this study, we introduce an innovative readout architecture
called Compression Shadow (CompShadow), which transforms the conventional
readout paradigm by compressing multi-qubit states into single-qubit shadows
before measurement. Compared to direct measurements of the initial quantum
states, CompShadow achieves comparable accuracy in amplitude and observable
expectation estimation while consuming similar measurement resources.
Furthermore, its implementation on near-term quantum hardware with nearest-
neighbor coupling architectures is straightforward. Significantly, CompShadow
brings forth novel features, including the complete suppression of correlated
readout noise, fundamentally reducing the quantum hardware demands for
readout. It also facilitates the exploration of multi-body system properties
through single-qubit probes and opens the door to designing quantum
communication protocols with exponential loss suppression. Our findings mark
the emergence of a new era in quantum state readout, setting the stage for a
revolutionary leap in quantum information processing capabilities.

In the realm of quantum physics, quantum measurement emerges as a pivotal and
distinctive element. Its paramount role resides in the capacity to extract
essential information from quantum states, enabling us to observe the
fascinating phenomena of quantum physics, including measurement-induced phase
transitions [1, 2, 3, 4]. Moreover, it serves as a prerequisite for harnessing
the quantum power, with phenomena like stochastic collapse forming the
security foundation for quantum key distribution [5, 6]. Thus, advancements in
measurement techniques are poised to revolutionize the development of quantum
information processing. For instance, classical shadow [7, 8, 9, 10, 11, 12,
13] methods have been proffered as a potent tool for extracting numerous
properties from quantum many-body systems with a significantly fewer
measurements. This innovation showcases the potential for more efficient
utilization of quantum resources in exploring complex quantum systems.
Additionally, the introduction of readout error mitigation methods [14, 15,
16, 17, 18, 19, 20] seeks to surmount inaccuracies in quantum state
information extraction caused by readout errors. These efforts result in a
substantial enhancement of the capabilities of near-term quantum computing
devices, marking a significant stride towards utility of quantum computing
before fault tolerance. A notable example is the successful simulation of a
127-qubit transverse-field Ising model achieved through error mitigation
techniques [21]. The confluence of these advancements underlines the depth of
impact that refined measurement techniques can have on the multifaceted
landscape of quantum information.

Here, we present a new paradigm for quantum state readout from a distinct
perspective, introducing the concept of quantum state compression shadow
(CompShadow). The primary goal is to compress the original quantum state into
a collection of single-qubit states while ensuring the faithful recovery of
the initial state’s information, including amplitudes or the expectation
values of specific observables, through measurement on these compressed
qubits. Importantly, CompShadow introduces a plethora of new functionalities,
augmenting quantum information processing capabilities: 1) It notably
addresses the challenging issue of correlated measurement errors in multi-
qubit systems. As quantum processors scaling up, the coupling and crosstalk
between qubits become significant, leading to a globally correlated error
channel with, in principle, an exponential cost for noise suppression [22, 23,
24, 25]. In contrast, CompShadow simplifies large-scale quantum state
measurements by measuring only a single-qubit, thereby elegantly removing
correlated errors. Numerical simulations reveal that CompShadow outperforms
mainstream readout error mitigation methods such as tensor product noise
inversion (TPN) [26], unfolding [23], and model-free method [27] on the near-
term quantum device, Zuchongzhi 2.1 [28, 29, 29]. Moreover, as gate accuracy
and qubit number increase, the advantage of CompShadow becomes more
pronounced. 2) CompShadow facilitates revealing the properties of multi-qubit
system using a single qubit prober, such as entanglement entropy and local
density, potentially playing a crucial role in many-body quantum simulations.
3) It also exhibits exponential resilience against loss in quantum state
transmission, significantly enhancing communication efficiency. These findings
underscore CompShadow as a novel paradigm for quantum state readout, poised to
markedly enhance the capabilities of contemporary quantum computing,
communication, and metrology.

Results

Framework

Compression shadows. Given a $n$-qubit quantum state $\rho$, CompShadows is a
collection of compressed quantum states
$\\{\sigma_{1},\sigma_{2},...,\sigma_{2^{n}-1}\\}$, obtained by applying a
series of compression circuits $U_{j}$ to $\rho$, retaining the first qubit,
and discarding the remaining qubits. Mathematically,

$\displaystyle\sigma_{j}=\text{Tr}_{\\{2,...,n\\}}(U_{j}\rho
U_{j}^{\dagger}),\ j=1,...,2^{n}-1,$ (1)

where $U_{j}$ represents a hardware-efficient shallow quantum circuit that
requires at most $2(n-1)$ nearest-neighbor two-qubit gates, composed as

$\displaystyle
U_{j}=\prod^{1}_{i=n-1}\text{CNOT}^{\beta_{ij}}(i+1,i)\text{CNOT}^{\alpha_{ij}}(i,i+1),$
(2)

where CNOT$(i,i+1)$ is an X gate on the $i+1$-th qubit controlled by the
$i$-th qubit. The exponents $\alpha,\beta$ of the gates are

$\displaystyle\alpha_{ij}$ $\displaystyle=\begin{cases}1,\quad j\geq 2^{i},\\\
0,\quad\text{else},\end{cases}$ (3) $\displaystyle\beta_{ij}$
$\displaystyle=\begin{cases}1,\quad j\geq 2^{i},\ j-2^{i}+1(\text{mod}\
2^{i})\leq 2^{i-1}\\\ 0,\quad\text{else}.\end{cases}$ (4)

The compression process, also shown in Fig. 1a, achieves a special Walsh
transform from the input $\rho$ to the CompShadows, as asserted in Theorem 1
(See Supplementary Note 1 for the proof). This connection enables us to
acquire information about $\rho$ by measuring the CompShadows, as outlined
below.

######  Theorem 1.

Let $A_{0}=1$, $A_{j}:=\text{Tr}(\sigma_{j}\ket{0}\bra{0})$ be the
$\ket{0}$-probability of the $j$-th CompShadow $\sigma_{j}$, and
$p_{i}:=\text{Tr}(\rho\ket{i}\bra{i})$ be the amplitude population of $\rho$,

$\displaystyle\mathbf{A}$ $\displaystyle=W\mathbf{p}$ (5)

in which $W=(H^{\otimes n}+E)/2$ is the 01-valued Walsh-Hadamard Transform
(WHT) matrix, Hadamard matrix $H=\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix}$ and
$E$ is the matrix of ones.

Amplitude population estimation. Using CompShadows, the full populations of
$\rho$ can be recovered, or a specific population. According to Eqn. (5),
obtaining the full amplitude vector $\mathbf{p}$ is straightforward, as

$\displaystyle\mathbf{p}$ $\displaystyle=W^{-1}\mathbf{A}$ (6)
$\displaystyle=\frac{1}{2^{n-1}}(H^{\otimes n}-2^{n-1}E_{00})\mathbf{A},$ (7)

in which $E_{00}=(\delta_{i0}\delta_{j0})_{ij}$ is the matrix where only the
element in the first row and first column is a non-zero one. Thus, by
performing only single-qubit measurements on CompShadows (see the upper half
of Fig. 1b), the population vector of $\rho$ can be recovered.

Recovering the full population vector demands all the $2^{n}-1$ shadows, while
estimating a single population $p_{a}$ for arbitrary $\ket{a}$ requires only a
polynomial number of shadows. Note that $p_{a}$ is the inner product between
the $a$-th row $(W^{-1})_{a,*}$ of inverse Walsh matrix $W^{-1}$ and the
shadow’s $\ket{0}$-probability vector $\mathbf{A}$. Borrowing concepts from
quantum-inspired techniques, specifically the Inner Product Estimation (IPE)
algorithm [30, 31, 32, 33], allows us to efficiently estimate the inner
product $p_{a}$ through subsampling on these vectors. For specific details,
refer to the Methods section. Here, we provide a concise overview of the
methodology. As illustrated in the lower-left corner of Fig. 1b, we randomly
sample indices $i_{1},i_{2},...,i_{r}$ according to a distribution about
$(W^{-1})_{a,*}$, where $r$ depends polynomially on the system size $n$ and
the tolerance error of the estimation. These indices then guide us to query
$(W^{-1})_{a,i_{1}},(W^{-1})_{a,i_{2}},...,(W^{-1})_{a,i_{r}}$ through direct
calculation, and $A_{a,i_{1}},A_{a,i_{2}},...,A_{a,i_{r}}$ by measuring the
corresponding shadows. The final inner product is obtained by calculating the
median of means of an unbiased estimator constructed from these elements.
Importantly, this approach allows us to efficiently estimate the inner product
$p_{a}$ for arbitrary $\ket{a}$. The required number of shadows, along with
the number of measurement shots, is detailed in the following theorem (See
Supplementary Note 2 for the proof).

######  Theorem 2.

The CompShadow population estimation method necessitates
$O(\frac{1}{\epsilon^{2}}\log(\frac{1}{\eta}))$ shadows and
$O(\frac{1}{\epsilon^{4}}\log(\frac{1}{\eta}))$ measurement shots to
approximate the population $p_{a}$ for any bitstring $\ket{a}$ within an error
margin of $\epsilon$ and a success probability of $1-\eta$.

Figure 1: The architecture and illustrative example applications of
compression shadow (CompShadow). (a) Generation of CompShadows. Utilizing
$2^{n}-1$ compression circuits on the $n$-qubit state $\rho$, $2^{n}-1$
independent CompShadows are produced. These circuits are composed of
information transmission modules $K_{ij}$ between $n-1$ pairs of adjacent
qubits. Each module consists solely of CNOT gates, of which the arrangements
vary in each circuit $U_{j}$, depending on the parameters $\alpha_{ij}$ and
$\beta_{ij}$ that we specifically configure in Equation (3,4). (b) The basic
functionalities of CompShadows. After measuring the $\ket{0}$ probabilities of
the CompShadows, the full populations can be obtained by applying an inverse
Walsh transform. Additionally, through the use of subsampling techniques [30,
31, 32, 33], it is possible to estimate individual populations with polynomial
shadows. Lastly, the estimation of observable expectations can be achieved
using just a single shadow. (c) The illustrative example applications of
CompShadow. Leveraging the unique characteristics of requiring measurement on
only a single-qubit, we can achieve high-fidelity quantum state readout,
completely unaffected by correlated readout errors. Furthermore, CompShadow
can serve as a single-qubit probe, allowing the exploration of properties in
multi-body quantum systems. Finally, the shadows can be used for quantum state
transmission, exhibiting strong resilience in the face of lossy channels.

Expectation estimation. Without loss of generality, we outline the method to
estimate $\braket{Z^{\otimes n}}$ for $\rho$ using CompShadow, since any
observable can be decomposed into a sum of Pauli observables, and these
observables can be further converted to $Z^{\otimes n}$ through single-qubit
rotations. The estimation of $\braket{Z^{\otimes n}}$ can be efficiently
accomplished using just one shadow and the same number of measurement shots as
a direct readout, where direct readout refers to the direct measurement of
$\rho$.

Based on Eqn. (6), we have

$\displaystyle\braket{Z^{\otimes n}}$ $\displaystyle=(\text{Diag}(Z^{\otimes
n}),\mathbf{p})$ (8) $\displaystyle=(1,-1)^{\otimes n}W^{-1}\mathbf{A}$
$\displaystyle=(-1,0,...,0,2)\mathbf{A}$ $\displaystyle=2A_{2^{n}-1}-1,$

which means $\braket{Z^{\otimes n}}$ of $\rho$ can be estimated by the
$\braket{Z}$ expectation of its $2^{n}-1$-th CompShadow, as

$\displaystyle{\text{Tr}[{\rho Z^{\otimes n}}]}$
$\displaystyle={\text{Tr}[{U_{2^{n}-1}\rho U_{2^{n}-1}^{\dagger}Z\otimes
I^{\otimes n-1}}]}$ $\displaystyle={\text{Tr}[{\sigma_{2^{n}-1}Z}]},$ (9)

where

$\displaystyle U_{2^{n}-1}$
$\displaystyle=\prod^{1}_{i=n-1}\text{CNOT}(i+1,i)$ (10)

is a shallow circuit with a depth of $n-1$, as illustrated in lower-right
corner of Fig. 1b. Theorem 3 states that the quantum state copy resources
required for expectation value estimation are the same for both CompShadow
readout and direct readout. Refer to its proof in Supplementary Note 2.

######  Theorem 3.

The CompShadow expectation estimation method requires
$O(\frac{1}{\epsilon^{2}})$ copies of the state to estimate the expectation of
any Pauli observable within an error of $\epsilon$.

Illustrative example applications

Figure 2: Comparative error analysis of four readout error mitigation methods.
The noise parameters are set based on typical values of Zuchongzhi 2.1 [28,
29]. (a) Estimation errors of the four methods (CompShadow, model-free, TPN,
and unfolding) with varying system size, given infinite measurement shots. The
CompShadow method typically exhibits the lowest estimation errors, and its
advantage becomes more pronounced with the growth of the system. (b)
Estimation errors of the four methods under different measurement shots. The
CompShadow method achieves the minimum estimation error with the fewest number
of measurements. (c) The estimation error ratio between the unfolding method
and the CompShadow method is depicted, considering amplified gate error
$\widetilde{\epsilon_{\text{G}}}=r_{G}\times\epsilon_{\text{G}}$ and
measurement error
$\widetilde{\epsilon_{\text{M}}}=r_{M}\times\epsilon_{\text{M}}$, where
$r_{G}$ ($r_{M}$) represents the amplification factor, and
$\epsilon_{\text{G}}$ ($\epsilon_{\text{M}}$) accounts for gate and readout
errors based on typical values from Zuchongzhi 2.1 [28, 29, 29]. The red curve
represents contours where both methods exhibit the same error. The shaded
region highlights the superiority of CompShadow over the unfolding method. (d)
The estimation error of CompShadow versus the reduction in gate error or the
amplification of measurement error, suggesting the performance of CompShadow
will continue to improve as the gate noise $\epsilon_{\text{G}}$ decreases,
and it remains steady with increasing measurement noise $\epsilon_{\text{M}}$.
This suggests that the performance of CompShadow will continue to improve as
the gate noise $\epsilon_{\text{G}}$ decreases and will not vary with an
increase in measurement noise $\epsilon_{\text{M}}$. All subplots depict the
average results obtained from 100 independent repeated experiments under
random computational basis states. The uncertainty bands around the curves
represent the standard error of the mean. Figure 3: CompShadow-probed
entanglement evolution of a six-qubit long-range XY model. The system
Hamiltonian is
$H=\hbar\sum_{i<j}J_{ij}(\sigma^{+}_{i}\sigma^{-}_{j}+\sigma^{-}_{i}\sigma^{+}_{j})+\hbar
B\sum_{j}\sigma^{z}_{j}$, where $\hbar=1$ and $J_{ij}=(j-i)^{-2}$. (a) The
entanglement entropies of states evolved under systems with and without random
disorder. The transverse field strength $B$ is fixed at 10 for the fixed field
or sampled randomly from a uniform distribution on [7,13] for the local
disorder. (b) The Rényi Mutual Information (RMI) of two pairs of subsystems
during the evolution of the Hamiltonian with disorder. In these subfigures,
the lines represent the theoretical values. The dots are averages of
CompShadow estimation results from 100 independent experiments. The error bars
on each dot represent the standard error of the mean. We find the probed value
of CompShadow faithfully reflect the evolution of entanglement in the many-
body system. Figure 4: The errors of direct communication and CompShadow
communication in the lossy channel. In each experiment, copies of GHZ states
are sent from Alice to Bob. Due to the presence of a lossy channel, the number
of received state copies by Bob decreases, introducing additional errors into
his observable estimation task. Simulations were conducted for different loss
rates (LR). (a) The errors curves with varying number of copies sent by Alice,
given a $9$-qubit state. (b) The errors curves with varying system size, given
$10000$ copies of a state sent by Alice. (c) Required copies of GHZ states
sent by Alice to achieve an estimation error less than $0.01$, varies with the
system size, considering a lossy channel with a loss rate of 0.3. We employ an
exponential function for the direct protocol and a linear function for the
CompShadow protocol to fit the curves. In these subfigures, the markers
indicate the average values from 100 independent experiments, while the bars
represent the standard errors of the means. It is evident that there is a
consistent exponential reduction in the required copy numbers for CompShadow
communication as the system size increases, compared to direct communication.

Correlation-noise-free readout. Qubit readout is generally the most error-
prone operation, degrading the performance of near-term quantum devices. While
fully suppressing readout noise usually requires exponential calibration
complexity due to the presence of correlated readout noise, CompShadow readout
naturally overcomes this issue, as CompShadow readout only requires single-
qubit readout. Nevertheless, CompShadow readout introduces additional quantum
circuit, referred to as the compression circuit, which brings extra noise.
Fortunately, we can apply randomized compiling (RC) [34, 35, 36, 37] to the
noisy compression circuits to mitigate the gate noises (see Methods for
detail). This approach reduces the circuit errors to an approximated Pauli
error channel while preserving the ideal functionality of the circuits.
Consequently, the noisy observable expectation will be nearly a constant
multiple of the ideal one.

Take the estimation of $\braket{Z^{\otimes n}}$ for example. Denote
$\sigma^{\prime}_{2^{n}-1}=\text{Tr}_{\\{2,...,n\\}}(\mathcal{U}_{\text{RC}}(\rho))$
as the ${2^{n}-1}$-th shadow of $\rho$, and
$\delta^{\prime}_{2^{n}-1}=\text{Tr}_{\\{2,...,n\\}}(\mathcal{U}_{\text{RC}}(\ket{0}\bra{0}^{\otimes
n}))$, where RC convert the compression circuit $U_{2^{n}-1}$ to channel
$\mathcal{U}_{\text{RC}}$. Then we estimate the error-mitigated observable
expectation as

$\displaystyle\frac{{\text{Tr}[{\sigma^{\prime}_{2^{n}-1}Z}]}}{{\text{Tr}[{\delta^{\prime}_{2^{n}-1}Z}]}}.$
(11)

.

To demonstrate the advantage of CompShadow readout over conventional readout
error mitigation methods, such as TPN [26], unfolding [23], and model-free
readout-error mitigation [27], we compare the estimation errors
$|\braket{Z^{\otimes n}}_{\text{estimated}}-\braket{Z^{\otimes
n}}_{\text{ideal}}|$ of these methods on random computational basis states,
where $\braket{Z^{\otimes n}}_{\text{estimated}}$ ($\braket{Z^{\otimes
n}}_{\text{ideal}}$) is the estimated (ideal) observable expectation. In our
simulation, we set the number of circuit instances for both CompShadow and
model-free to $4n$. The single-qubit and two-qubit gate errors, as well as the
$T_{1}$, are based on typical values of Zuchongzhi 2.1 [28, 29] and are set at
0.16%, 0.6%, and $26.5\mu\text{s}$, respectively. The readout errors arise
from our calibration data (see Supplementary Note 3), which includes
correlated errors in the error matrix, with an average error denoted as 2.22%.

Figure 2 illustrates the simulation results. In Fig. 2a, we examine the
relationship between estimation errors and the system size $n$. The number of
measurement shots is considered infinite to avoid the influence of statistical
errors. As anticipated, the estimation errors for all four methods increase as
the system size grows. Notably, as the system size expands, the superiority of
CompShadow readout becomes increasingly evident among the four methods. For
instance, for $n=6$, the errors of CompShadow readout are
$55.3\%,32.3\%,18.5\%$ of those for unfolding, TPN, and model-free,
respectively. The performance of the model-free method significantly degrades
with the increasing system size, as the number of circuit instances we employ
($4n$) is much lower than the theoretically required quantity for the model-
free method to function optimally, which is $2^{n}$. In Fig. 2b, we maintain
the system size at $n=6$ and vary the number of measurement shots. The errors
for all methods decrease with the number of measurement shots and gradually
converge. Notably, CS requires the fewest number of measurement shots to
achieve low estimation error, and it converges most rapidly. Figure 2a,b
underscore the evident advantage of CS over other methods in mitigating
measurement errors and its robust scalability concerning the number of qubits
and measurement shots.

Next, we fix the system size at $n=6$ and investigate the performance of
CompShadow readout under higher and lower noise scenarios, with infinite
measurement shots. In Fig. 2c, we systematically amplify gate error
$\epsilon_{G}$ and readout error $\epsilon_{M}$. The contour plot illustrates
the ratio of estimation errors between the unfolding and CompShadow methods
(two methods for the victory in Fig. 2a,b.), with contours of ratio $1$
highlighted in red. We observe a substantial advantage of CompShadow readout
in scenarios with high measurement error and low gate error. Remarkably, even
as gate error $\epsilon_{\text{G}}$ is magnified to three times large, as long
as there is a slight increase in measurement error $\epsilon_{\text{M}}$ (to
about $50\%$ larger), the advantage of CompShadow readout persists. Therefore,
CompShadow readout can serve as a valuable complement for readout-error
mitigation methods, particularly in systems characterized by high readout
errors and low gate errors. In Fig. 2d, we fix measurement error
$\epsilon_{\text{M}}$, revealing that the estimation error of CompShadow
readout decreases polynomially with gate error $\epsilon_{\text{G}}$. When the
gate error is reduced to $10^{-6}$ of its original level, the estimation error
of CompShadow can be minimized to about $8\times 10^{-9}$. Conversely, when
fixing gate error $\epsilon_{\text{G}}$ and increasing measurement error
$\epsilon_{\text{M}}$, the estimation errors of CompShadow for all cases are
approximately 0.008, with some minor fluctuations mainly attributed to the
randomness in the RC circuit.

Single-qubit probe for many-body systems. The application of CompShadow
readout for probing the properties of many-body systems not only effectively
mitigates readout errors, as discussed earlier, but also introduces a
distinctive feature. The compressed individual qubit, referred to as a single-
qubit probe, can either be an integral part of the system or an auxiliary
qubit derived from another system (see the Supplementary Note 5). This feature
allows for a scenario where a particular many-body system is physically
challenging to measure, we can leverage an easily measurable system,
implementing the CompShadow compression circuit, to interact with it. By
reading the easily measurable system, we can gain insights into the properties
of the target system. In this context, we present several examples, such as
utilizing CS to measure second-order Rényi entropy and Rényi mutual
information (RMI), defined as

$\displaystyle S^{(2)}(\rho)$ $\displaystyle=-\log_{2}\text{Tr}(\rho^{2}),$
(12) $\displaystyle I^{(2)}(\rho_{A}:\rho_{B})$
$\displaystyle=S^{(2)}(\rho_{A})+S^{(2)}(\rho_{B})-S^{(2)}(\rho_{AB}),$ (13)

and local density [38, 39] (refer to the Supplementary Note 4 for specific
details).

Combining CompShadow readout with the randomized measurement (RM) methods [40,
10] allows for the single-qubit probing of the second-order Rényi entropy. The
main idea is that given copies of the state $\rho$, for $M$ rounds, we apply a
layer of random single-qubit Clifford gates on the system $\rho$. The outcome
states, denoted as $U^{(i)}\rho U^{(i)\dagger},i=1,...,M$, are then compressed
into shadows and measured. For a more detailed process, see the Methods.

Numerical simulations are conducted on a 6-qubit long-range XY model [41]
defined by the Hamiltonian
$H=\hbar\sum_{i<j}J_{ij}(\sigma^{+}_{i}\sigma^{-}_{j}+\sigma^{-}_{i}\sigma^{+}_{j})+\hbar
B\sum_{j}\sigma^{z}_{j}$, where $\hbar=1$ and $J_{ij}=(j-i)^{-2}$. The
transverse field is fixed at $B=10$, and local disorder potentials $B_{j}$ are
uniformly sampled from the interval $[7,13]$. In Fig. 3a, the evolution of
entanglement entropies with and without random disorder is presented. The
entanglement entropy of both systems initially undergoes a rapid increase,
followed by sustained oscillations. The system without disorder generally
exhibits a higher degree of entanglement. Fig. 3b illustrates the RMI for two
pairs of subsystems during the evolution of the perturbed Hamiltonian. The
distant subsystems display a lower degree of RMI compared to the adjacent
ones. In both cases, the CompShadow method (dots) generally align with the
theoretical predictions (lines). These results showcase the capability of
CompShadow in measuring the entanglement structure of many-body systems.

Loss-resilient quantum state transmission. Due to CompShadow compressing
quantum states into a single qubit, transmitting the state after CompShadow
compression through a lossy channel is more robust against errors compared to
transmitting the original state. This characteristic bears significant
implications for quantum communication or applications in a quantum internet.

Here, we present a proof-of-concept example. Let’s consider Alice transmitting
an $n$-qubit GHZ state to Bob, who aims to estimate the expectation value of
$Z^{\otimes n}$. Alice has two options: she can directly send the $n$ qubits
(referred to as ”direct communication”), or she can compress the quantum state
into a single-qubit shadow using the compression circuit $U_{2^{n}-1}$ in Eqn.
(10) (referred to as ”CompShadow communication”). In both scenarios, the loss
of a single qubit is considered a transmission failure. In the transmission
channel, each qubit faces an independent and fixed loss rate.

In Fig. 4a,b, we fix the system size and the number of copies of the quantum
state transmitted by Alice, and then vary the number of copies and system
size, observing the estimation error of CompShadow communication and direct
communication. Notably, due to the compression into a single qubit, CompShadow
demonstrates a marked advantage against lossy errors. The estimation error in
the CompShadow protocol is significantly lower, and this advantage becomes
more pronounced with decreasing copy numbers and increasing system sizes. The
primary reason for this lies in the CompShadow protocol, where Bob obtains
copies ($(1-r)n_{c}$) with an exponential growth rate compared to direct
communication ($(1-r)^{n_{q}}n_{c}$), where $r$ is the loss rate, $n_{q}$ and
$n_{c}$ is the system size and number of copies Alice sent. Finally, in Fig.
4c, we set the target to achieve an estimation error less than $0.01$ and
compare the minimum number of copies required for both protocols under
different system sizes. As anticipated, the CompShadow protocol requires
exponential fewer copies compared to the direct protocol with the system size
increasing. When the system size reaches $n=11$, direct communication requires
approximately six times more number of copies than CompShadow communication.
This series of experiments clearly illustrates the advantages of the
CompShadow protocol in transmitting quantum states through lossy channels, not
only in reducing copy numbers but also in improving measurement accuracy.

Discussion

The CampShadow efficiently compresses a large-scale quantum state into
individual qubits, introducing only some additional hardware-efficient shallow
quantum circuits with at most $2(n-1)$ nearest-neighbor two- qubit gates for
$n$ qubits. This approach is highly practical for near-term quantum devices.
The compression process can be conceptualized as a lossless quantum auto-
encoder, as we can recover information such as amplitudes and observable
expectations of the original quantum state through measurements on the
compressed single qubit. The characteristics of the CampShadow provides a
robust impetus for the advancement of quantum technologies, such as improved
readout accuracy and an exponential suppression of losses in quantum state
transmission. Besides, we would like to share some additional insights:

Firstly, due to the fact that CampShadow readout only requires measurements on
individual qubits, errors in single-qubit measurements can be nearly
completely eliminated using the TPN error mitigation method. This may imply a
new direction in hardware development, where the focus can shift towards
enhancing the fidelity of quantum gates without the need to simultaneously
optimize multi-qubit readout performance. This shift could potentially
significantly reduce the complexity of designing and manufacturing high-
performance hardware.

Secondly, CompShadow is a versatile framework with strong flexibility. In
Supplementary Note 5, we demonstrate that the compressed qubit can exist
either within the system or as an external ancillary qubit. Additionally, the
number of qubits after compression can be adjusted to reduce the depth of the
compression circuit.

Thirdly, the application of the single-qubit prober can be considered a form
of weak measurement. The unmeasured portion of the system still contains
information about the original system, potentially exploitable to unveil more
quantum properties.

Finally, the CompShadow readout involves a Walsh transform from quantum states
to classical vectors. Considering the wide-ranging applications of Walsh
transforms in fields like image compression, signal processing, and
cryptography, the compression circuit of our method has the potential to
catalyze the quantization of related applications.

Methods

Amplitude population estimation

We present the method for estimating an amplitude population $p_{a}$,
utilizing only a polynomial number of shadows. Note that
$p_{a}=((W^{-1})_{a,*},\mathbf{A})$ is a specific inner product between
vectors of exponential length. The avoidance of exponential calculation
complexity is accomplished through the inner product estimation method (IPE)
[30, 31, 32, 33]. Generally, to estimate the inner product between vector $A$
and $B$, the method repeatedly samples the length-square distribution
$\mathbb{P}(i)=\frac{|A_{i}|^{2}}{\|A\|^{2}}$ of $A$, generating indices
$i_{1},i_{2},...,i_{r}$. Subsequently, a median of means for the values
$\frac{\|A\|^{2}}{A_{i}}B_{i},i=i_{1},i_{2},...,i_{r}$ is calculated as the
estimation of $(A,B)$. The pseudo code in Alg. 1 outlines the parameter choice
for IPE.

1:A sampling oracle and a query oracle for $A\in\mathbb{C}^{N}$. A query
oracle for $B\in\mathbb{C}^{N}$. The Frobenius norm squared $\|A\|_{F}^{2}$.
Relative error bound $\xi$ and success probability bound $1-\eta$.

2:Estimate $(A,B)$.

3:Sample $A$ for $s=\lceil\frac{9}{\xi^{2}}\rceil$ times, yielding
$i_{1},...,i_{s}$. Calculate the mean of $\frac{\|A\|^{2}}{A_{i}}B_{i}$ among
these samples.

4:Repeat Step 1 for $\lceil 6\log_{2}(\frac{2}{\eta})\rceil$ times and
calculate the median of the $s$ outputs, denoted as $x$.

5:$x$.

Algorithm 1 Inner Product Estimation.

The following Lemma 1 states the complexity of the IPE algorithm. See its
proof in Ref. [31, 30].

######  Lemma 1.

Suppose that we have sampling access to $A\in\mathbb{C}^{N}$ in complexity
$L(A)$ and query access to $B\in\mathbb{C}^{N}$ in complexity $Q(B)$. Then we
can estimate $(A,B)$ to precision $\xi\|A\|\|B\|$ with probability at least
$1-\eta$ in time

$\displaystyle O\left(\frac{\log(1/\eta)}{\xi^{2}}(L(A)+Q(B))\right).$ (14)

In our case, vector $A$ corresponds to the $a$-th row of the inverse Walsh
matrix $W^{-1}$, and vector $B$ represents the $\ket{0}$-probabilities
$\mathbf{A}$ of the shadows. Referring to the input in Alg. 1, it is essential
to clarify how to sample the length-square distribution of $(W^{-1})_{a,*}$,
query the elements of $\mathbf{A}$, and define the norm squared
$\|(W^{-1})_{a,*}\|^{2}$. To begin, the element$A_{j}$ are queried by
measuring the $j$-th shadow. Subsequently, leveraging on our knowledge about
Walsh matrix, we can easily find

$\displaystyle(W^{-1})^{2}_{ij}$
$\displaystyle=\begin{cases}(2^{1-n}-1)^{2},&i=j=0,\\\
2^{2-2n},&\text{else},\end{cases}$ (15) $\displaystyle\|(W^{-1})_{i,*}\|^{2}$
$\displaystyle=\begin{cases}1,&i=0,\\\ 2^{2-n},&\text{else}.\end{cases}$ (16)

Finally, we formulate an efficient length-square sampling algorithm for
$(W^{-1})_{a,*}$ with arbitrary value of $a$, shown in Alg. 2. To achieve
amplitude population estimation within a precision of $\epsilon$ and a success
rate of $1-\eta$, the IPE parameters are taken as $\xi=\epsilon/2$ and $\eta$.
According to Lemma 1, the time complexity for executing IPE is
$O(\frac{1}{\epsilon^{4}}\log(\frac{1}{\eta}))$. A detailed complexity
analysis of the amplitude population estimation algorithm is presented in the
proof of Theorem 2 in Supplementary Note 2.

1:System size $n$. Row index $i$.

2:Sampling an index $j$ that follows the distribution
$\mathbb{P}(j)=\frac{(W^{-1})^{2}_{ij}}{\|(W^{-1})_{i,*}\|^{2}}$.

3:If $i\neq 0$, sample $j$ from a uniform distribution in the set
$\\{0,1,...,2^{n}-1\\}$. Output $j$ and end the algorithm.

4:If $i=0$, generate a random real number $x$ from uniform distribution
$[0,1]$. If $x\leq(2^{1-n}-1)^{2}$, return $j=0$. Otherwise, sample $j$ from a
uniform distribution in the set $\\{1,...,2^{n}-1\\}$.

5:$j$.

Algorithm 2 Sampling a certain row of inverse Walsh matrix.

Simplifying errors in the compression circuit

To simplify the errors, including circuit errors and the single-qubit
measurement errors, into a single Pauli error channel, we employ randomized
compiling (RC) across the noisy compression circuit. As depicted in Fig. 1c,
before the compression circuit $U$, we randomly select a layer of Pauli gates
$\otimes_{i=1}^{n}P^{(i)}$ to apply to the target state. After the
measurements, we classically conduct $U^{\dagger}\otimes_{i=1}^{n}P^{(i)}U$,
which, due to the Cliffordness of $U$, is also a layer of Pauli gates
$\otimes_{i=1}^{n}Q^{(i)}$ to the measurement outcomes. As we only measure the
first qubit, the Paulis on other qubits can be omitted. The remaining
operation becomes: If $Q^{(0)}$ equals $X$ or $Y$, we flip the measured bits;
otherwise, we leave it unchanged. This process is repeated in multiple rounds,
the average of the obtained measurement results is taken.

Next we analyze the impact of RC on CompShadow readout. Without loss of
generality, we focus on the estimation of observable expectation $Z^{\otimes
n}$. Let $\tilde{\mathcal{U}}$ denote the channel of the noisy compression
circuit, $\mathcal{E}_{m}$ represent the measurement error channel on the
first qubit, $\mathcal{I}$ stand for the identity channel on each qubit. Then,
the noisy expectation value of $Z^{\otimes n}$ is

$\displaystyle\braket{Z^{\otimes
n}}_{\text{noisy}}={\text{Tr}[{\mathcal{E}_{m}\otimes\mathcal{I}^{\otimes
n-1}\circ\tilde{\mathcal{U}}(\rho)Z\otimes I^{\otimes n-1}}]},$

Let $\mathcal{E}_{c}(\cdot)=\tilde{\mathcal{U}}(U^{\dagger}\cdot U)$, the
channel $\mathcal{E}=\mathcal{E}_{m}\otimes\mathcal{I}^{\otimes
n-1}\circ\mathcal{E}_{c}$ is then the overall noise channel. As described
above, in the $j$-th round, we apply the Pauli gates
$\otimes_{i=1}^{n}P^{(i)}$ on $\rho$, and apply
$U^{\dagger}\otimes_{i=1}^{n}P^{(i)}U$ on the measurement outcomes. If we
iterate through all $n$-qubit Pauli operators with $4^{n}$ rounds, the error
channel $\mathcal{E}$ is converted to a Pauli noise channel
$\mathcal{E}_{p}(\cdot)=\sum_{i}\lambda_{i}P_{i}\cdot P_{i}$, in which $P_{i}$
are $n$-qubit Pauli operators [42, 37]. In this case,

$\displaystyle\braket{Z^{\otimes n}}_{\text{twirl}}$
$\displaystyle={\text{Tr}[{\mathcal{E}_{p}(U\rho U^{\dagger})Z\otimes
I^{\otimes n-1}}]}$ $\displaystyle=\sum_{i}\lambda_{i}{\text{Tr}[{P_{i}U\rho
U^{\dagger}P_{i}Z\otimes I^{\otimes n-1}}]}$
$\displaystyle=\sum_{i}\lambda_{i}{\text{Tr}[{U\rho
U^{\dagger}P_{i}^{(1)}ZP_{i}^{(1)}\otimes I^{\otimes n-1}}]}$
$\displaystyle=\sum_{i}\lambda_{i}h_{i}{\text{Tr}[{U\rho U^{\dagger}Z\otimes
I^{\otimes n-1}}]}$ $\displaystyle=\sum_{i}\lambda_{i}h_{i}\braket{Z^{\otimes
n}}_{\text{ideal}},$ (17)

in which $P_{i}^{(1)}$ is the Pauli operator on the first qubit of $P_{i}$,
and

$\displaystyle h_{i}=\begin{cases}1,&P_{i}^{(1)}=I,Z,\\\
-1,&P_{i}^{(1)}=X,Y.\end{cases}$

Eqn. (17) suggests that the noisy expectation is a fixed multiple of the ideal
one if all $4^{n}$ Pauli operators are iterated. The multiplier
$\sum_{i}\lambda_{i}h_{i}$ can be estimated by measuring the $Z^{\otimes n}$
expectation for state $\ket{0}\bra{0}^{\otimes n}$ through the compression
circuit, leading to the error mitigation formula as Eqn. (11). In our
experiments, we typically employ $4n$ random selected Pauli operators, which
approximately achieve this goal. The choice of the linear number $4n$ is
efficientand empirically effective, as demonstrated in Supplementary Note 3.

Entanglement entropy probing

We first derive a calculation formula suitable for CompShadow. Drawing from
the randomized measurement (RM) methods [40, 10], the second-order Rényi
entropy of a quantum state $\rho$ can be estimated by

$S^{(2)}(\rho)=-\log_{2}\frac{2^{n}}{M}\sum_{i}\sum_{s,s^{\prime}}(-2)^{-h(s,s^{\prime})}\mathbb{P}(s,i)\mathbb{P}(s^{\prime},i),$
(18)

in which $s,s^{\prime}$ are summed over all measured bitstrings,
$h(s,s^{\prime})$ is the Hamming distance between $s$ and $s^{\prime}$, and
$\mathbb{P}(s,i)$ is the $s$-th amplitude population of state $U^{(i)}\rho
U^{(i)\dagger}$.

Eqn. (18) can be rewritten as

$S^{(2)}(\rho)=-\log_{2}\frac{1}{M}\sum_{i}\mathbf{p}^{(i)T}D\mathbf{p}^{(i)},$
(19)

in which $\mathbf{p}^{(i)}$ is the amplitude population vector of $U^{(i)}\rho
U^{(i)\dagger}$ in $i$-th round, and matrix $D_{ij}=2^{n}(-2)^{-h(i,j)}$. By
Eqn. (6), we have

$S^{(2)}(\rho)=-\log_{2}\frac{1}{M}\sum_{i}\mathbf{A}^{(i)T}W^{-T}DW^{-1}\mathbf{A}^{(i)},$
(20)

in which

$W^{-T}DW^{-1}=\begin{pmatrix}2^{n}&-\frac{d_{1}}{2}&-\frac{d_{2}}{2}&\cdots&-\frac{d_{2^{n}-1}}{2}\\\
-\frac{d_{1}}{2}&d_{1}&0&\cdots&0\\\ -\frac{d_{2}}{2}&0&d_{2}&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
-\frac{d_{2^{n}-1}}{2}&0&0&\ldots&d_{2^{n}-1}\end{pmatrix},$

and

$\displaystyle d_{j}=3^{h(j,0)}2^{2-n}.$ (21)

Therefore,

$S^{(2)}(\rho)=-\log_{2}\frac{1}{M}\sum_{i}\left(2^{n}+\sum_{j=1}^{2^{n}-1}d_{j}(A^{(i)2}_{j}-A^{(i)}_{j})\right),$
(22)

Then, we utilize the IPE algorithm (Alg. 1) to efficiently determine the sum
$\sum_{j=1}^{2^{n}-1}d_{j}(A^{(i)2}_{j}-A^{(i)}_{j})$. For the input of IPE,
the sampling oracle for $\mathbf{d}$ is provided in Alg. 3. The Frobenius norm
can be found as $\|\mathbf{d}\|^{2}_{F}=2^{4-2n}(10^{n}-1)$. To query the
element $A^{(i)2}_{j}-A^{(i)}_{j}$, we note the measured value of
$A^{(i)}_{j}$ can not be directly substituted into $A^{(i)2}_{j}-A^{(i)}_{j}$,
because $\mathbb{E}\tilde{A}^{2}\neq(\mathbb{E}\tilde{A})^{2}$. An unbiased
estimator can be designed as

$\widetilde{A^{(i)2}_{j}-A^{(i)}_{j}}=\frac{1}{N(N-1)}\sum_{k\neq
l}(1-X_{k})X_{l},$ (23)

in which $X_{1},...,X_{N}$ are the measurement outcomes from the shadow
$\sigma_{j}^{(i)}$.

1:System size $n$.

2:Sampling an index $j$ that follows the distribution
$\mathbb{P}({j})=\frac{d_{j}^{2}}{\|\mathbf{d}\|^{2}}=\frac{3^{2h(j,0)}}{10^{n}-1}$.

3:Generate $n$ random real number $x_{1},...,x_{n}$ from uniform distribution
$[0,1]$.

4:For $k=1,...,n$, if $x_{k}<0.1$, $j_{k}=0$, else $j_{k}=1$. Yield a $n$-bit
binary number $j=(j_{n}j_{n-1}...j_{1})_{2}$ accordingly.

5:If $j=(0...0)_{2}$, go to Step 1. Otherwise, return $j$.

6:$j$.

Algorithm 3 Efficient sampling of vector $\mathbf{d}$.

Competing Interests

The authors declare no competing interests.

Author Contributions H.-L. H. supervised the whole project. C. D. and H.-L. H.
conceived the idea and co-wrote the paper. C. D. provided the theoretical
analysis and proofs. C. D., X.-Y. X. and H.-L. H. carried out the numerical
simulations and analyzed the results. All authors contributed to discussions
of the results.

Acknowledgments

H.-L. H. is supported by the Youth Talent Lifting Project (Grant No.
2020-JCJQ-QT-030), National Natural Science Foundation of China (Grants No.
12274464).

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