# Quantum computing hardware in the cloud:
Should a computational chemist care?

Alessandro Rossi<EMAIL_ADDRESS>Department of Physics, SUPA,
University of Strathclyde, Glasgow G4 0NG, United Kingdom  National Physical
Laboratory, Hampton Road, Teddington TW11 0LW, United Kingdom  Paul G. Baity
James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT,
United Kingdom  Vera M. Schäfer  Department of Physics, University of Oxford,
Clarendon Laboratory, Parks Road, Oxford OX1 3PU, United Kingdom  Martin
Weides  James Watt School of Engineering, University of Glasgow, Glasgow G12
8LT, United Kingdom

###### Abstract

Within the last decade much progress has been made in the experimental
realisation of quantum computing hardware based on a variety of physical
systems. Rapid progress has been fuelled by the conviction that sufficiently
powerful quantum machines will herald enormous computational advantages in
many fields, including chemical research. A quantum computer capable of
simulating the electronic structures of complex molecules would be a game
changer for the design of new drugs and materials. Given the potential
implications of this technology, there is a need within the chemistry
community to keep abreast with the latest developments as well as becoming
involved in experimentation with quantum prototypes. To facilitate this, here
we review the types of quantum computing hardware that have been made
available to the public through cloud services. We focus on three
architectures, namely superconductors, trapped ions and semiconductors. For
each one we summarise the basic physical operations, requirements and
performance. We discuss to what extent each system has been used for molecular
chemistry problems and highlight the most pressing hardware issues to be
solved for a chemistry-relevant quantum advantage to eventually emerge.

††preprint: APS/123-QED

##  I Introduction

This year marks exactly 40 years since Richard Feynman famously said [1]:
“Nature isn’t classical, dammit, and if you want to make a simulation of
nature, you’d better make it quantum mechanical, and by golly it’s a wonderful
problem, because it doesn’t look so easy”. On the one hand, the visionary
physicist anticipated the possibility (and the inherent difficulty) of
building a new type of computing apparatus operating according to the laws of
quantum mechanics. On the other hand, he had immediately identified one of its
most useful areas of application, i.e. simulations of chemical and physical
systems.
Computational chemists will indeed benefit from future quantum computers for
calculations of molecular energies to within chemical accuracy, defined to be
the target accuracy necessary to estimate chemical reaction rates at room
temperature ( $\approx 1$ kcal/mol) [2]. Fully-fledged, error-free quantum
systems will enable predictions and simulations that are not possible today in
terms of both accuracy and speed. This could have a revolutionary impact on
the design of drugs, catalysts and materials by allowing computational methods
to replace lengthy and expensive experimental procedures. Unfortunately, we
are still in the infancy of the development of quantum computing technology
and a machine that provides a quantum advantage in molecular chemistry over
classical super-computers has not emerged yet. However, the progress in
handling increasingly complex molecular and material chemistry has been
relentless. Small-scale quantum machines developed by academic or corporate
research centres have been initially used to simulate simple diatomic or
triatomic molecules made up of just H and He atoms [3, 4, 5]. Recently, more
powerful quantum computers have been used to simulate larger compounds
containing N, Li and Be atoms [6, 7, 8]. Although these studies do not show a
clear advantage in using quantum computing over the conventional computational
methods that have been used for their validation, they do indicate that
hurdles are being tackled and viable ways forward are becoming available.
The major impediments that currently stifle quantum computers are limits to
the number of computational units and computational errors. The units of
quantum information are called quantum bits (qubits) in analogy with the
binary bits of classical computers. Quantum algorithms for chemical
calculations use qubit-based Hamiltonians to map molecular many-body
Hamiltonians and evaluate the system wavefunction through repeated sampling of
the qubit register states [9, 10, 11]. One particular algorithm, namely the
variational quantum eigensolver (VQE) [12], has acquired prominence because it
alleviates the computational burden on today’s limited quantum machines by
using a classical co-processor to support the calculation. To date, the most
advanced VQE simulations have mapped just 24 molecular orbitals onto 12 qubits
[6], a relatively easy feat for traditional computers. In order to calculate
the energy ground state of more complex systems with chemical accuracy, it is
expected that the number of qubits available will need to increase by orders
of magnitude. A recent estimate [2] indicates that more than 1500 orbitals are
required for a VQE calculation that could outperform classical super
computers.
The other hurdle to consider is that qubits are error-prone due to noise-
limited phase coherence, with inherent challenges in reading and writing their
states properly (qubit fidelity). Ultimately, there is a limit to the number
and duration of operations (qubit gates) that a quantum computer can carry out
before error propagation leads to computational failure. Quantum error
correction (QEC) schemes to correct these errors have been identified [13,
14]. The main drawback is that QEC leads to hardware aggravation, given that
several physical qubits are required to realise a single error-corrected
“logical” qubit. Some estimates based on realistic qubit noise levels conclude
that the ratio of physical to logical qubits to reach fault tolerant machines
could be at least $1$,$000:1$ [15].
It is, therefore, evident that, to approach quantum chemistry simulations in a
meaningful way, quantum computers with millions of physical qubits will be
required, if one has to accurately map thousands of spin-orbitals. By
contrast, today’s quantum computers rely on a small number of noisy qubits
(less than 100 at present) because the ability to manufacture, interconnect
and error-correct qubits on larger scales is not yet sufficiently developed.
This is why quantum machines are presently dubbed NISQ (Noisy Intermediate-
Scale Quantum) [16]. An important figure of merit for NISQ systems is called
quantum volume (QV) [17], which combines in one convenient metric the number
of qubits available, how extensively they are interconnected, and their gate
fidelity. A larger QV indicates that more complex quantum algorithms can be
successfully run. This metric clearly shows that, to increase the
computational power, is not sufficient to build machines with more qubits if
these remain affected by high levels of noise. Hence, the challenge of
improving quantum computing power is a coordinated effort in scaling up
qubits, making them as interconnected as possible, and reducing the error
rates.
NISQ computers come in a variety of hardware implementations. Different from
classical computers for which the Central Processing Unit (CPU) is invariably
made with silicon integrated technology, Quantum Processing Units (QPU) can
also be realised with superconductor microchips, ions or neutral atoms trapped
in a vacuum, and on-chip photonic waveguides. Different technologies present
different trade-offs in terms of number of qubits, phase coherence time, qubit
fidelity, connectivity etc. Here, we are going to focus on a specific subset
of quantum hardware types. Specifically, we will look at digital programmable
QPUs, as opposed to adiabatic or analog systems [18, 19]. Among these, we
shall discuss computers available to the general public through cloud
services. On the one hand, being available to the public, and not just to
specialised quantum developers, indicates these systems have reached superior
maturity. On the other hand, we feel that a description of how these systems
operate at hardware level will benefit the reader who may have to navigate
through offers and subscription packages to identify the most relevant service
for the computational chemistry application of interest. This may indeed
become a daunting task without prior knowledge given the pace with which these
services are becoming available and compete to acquire large customer bases.
Global corporations offering cloud access include Google, IBM, Microsoft and
Amazon. We shall limit our discussion to three types of hardware in the cloud:
superconductor-, ion trap- and silicon-based quantum computers. For each one
of these systems we discuss how qubits are physically embodied, initialised,
read and manipulated. We will describe the operational requirements and the
main performance parameters of each implementation. We will provide some use
cases relevant to quantum chemical simulations to exemplify the usefulness of
different machines in relevant contexts. The remainder of this Article is
organised as follows. Superconductor devices are described in Section II, ion
trap systems in Section III, and a silicon processor in Section IV. These
technologies are compared in Section V, and finally an outlook for future
developments is discussed in Section VI.

##  II Superconducting quantum computers

Superconducting (SC) circuits are the most widely used systems for quantum
computing. Many industry leaders, such as Google, IBM, and Rigetti, use
superconducting quantum circuits to realize their quantum computers. Qubits
implemented on superconducting devices fulfil the requirements [20] for
scalable quantum computing, and therefore micron-sized quantum circuits and
associated integrated-circuit processing techniques can be scaled up when
implemented using superconducting quantum technologies. Whereas trapped ion
and silicon devices control and read (sub-)atom scale components as their
quantum systems, in SC circuits information is encoded into a _macroscopic_
quantum state of the condensate of paired electrons (so-called Cooper pairs),
which collectively participate in a charged superfluid state with a wave
function $\Psi(\vec{r},t)=|\Psi(\vec{r},t)|e^{i\phi(\vec{r},t)}$ [21, 22].
Here, the wave function parameters $|\Psi(\vec{r},t)|^{2}$ and
$\phi(\vec{r},t)$ describe the density of Cooper pairs and complex phase of
the condensate as a function of position $\vec{r}$ and time $t$.
Superconducting qubits, such as the one shown in Fig. 1(a), consist of islands
of superconducting material, such as aluminium, connected by one or more
Josephson junctions [22], which are nm-thin insulating barriers made from e.g.
aluminium oxide.

Figure 1: (a) Concentric transmon qubit design from Ref. [23] and (inset) its
equivalent circuit diagram. Two superconducting islands (green and blue) are
shunted by a small Josephson junction bridge (orange). The qubit state is read
out using a coplanar waveguide resonator (red). This readout resonator is
inductively coupled to a signal line (black). (b) The states of the transmon
qubit are determined by the sinusoidal potential (black solid line) of the
Josephson junction. Solved in the phase basis ($\Delta\phi$), the Eigen
energies (solid colored lines) can be approximated by a harmonic oscillator
(dashed lines, respective colors) whose degeneracies are lifted by first order
corrections from capacitive charging energy on the junction [24, 25, 26]. (c)
Diagram of the Bloch sphere. The ground $\ket{0}$ and first excited $\ket{1}$
states are used to define the qubit’s logical state $\ket{\psi}$, which is a
linear combination of $\ket{0}$ and $\ket{1}$ with respective complex
amplitudes $\alpha$ and $\beta$. $\ket{\psi}$ can be manipulated by voltage
pulses and gating operations and read out by projection onto a specified
measurement basis.

The current $I$ passing through the Josephson junction depends on the phase
difference $\Delta\phi$ between the superconductors at either side of the
junction by the relation $I=I_{0}\sin(\Delta\phi)$, where $I_{0}$ is the
largest supercurrent supported by the junction. When a voltage difference $V$
occurs across the junction, $\Delta\phi$ changes as
$\frac{d\Delta\phi}{dt}=2eV/\hbar$ [27, 22], where $e$ is the electron charge
and $\hbar$ is the reduced Planck’s constant. This time dependence leads to
non-linear resonance behavior with quantized states that are determined by
flux, charge, and phase degrees of freedom [28].

The effective circuit diagram of a superconducting qubit is shown in the inset
of Fig. 1(a) and can be described by the Hamiltonian [24, 26]

$H=4E_{C}(\Delta n)^{2}-E_{J}\cos(\Delta\phi),$ (1)

where $E_{J}$ is the energy of the current passing through the junction and
$E_{C}$ is the capacitive charging energy between the two superconducting
islands. Quantum states are usually determined in either the basis of
$\Delta\phi$, as shown in Fig. 1(b), or capacitive charge number $\Delta n$,
depending on the relative strengths of $E_{J}$ and $E_{C}$. Similar to the
conjugate variables of position and momentum, $\Delta\phi$ and $\Delta n$ have
a non-zero commutation $[\Delta\phi,\Delta n]=i$ [26] and uncertainty relation
$\sigma_{\Delta\phi}\sigma_{\Delta n}\gtrsim 1$ [22]. In either basis, quantum
states are approximated by a harmonic oscillator in which degeneracy is lifted
by the non-dominant energy term. This degeneracy allows for the
differentiation of qubit states. Superconducting qubits are typically operated
at the transition between the ground $\ket{0}$ and first excited $\ket{1}$
states.

One of the benefits of superconducting qubits is the ability to engineer a
wide range of operational parameters by tuning the parameters $E_{J}$ and
$E_{C}$ through intentional design choices. Perhaps the most widely used
design choice is to have $E_{J}/E_{C}\sim 10^{2}$. This is the so-called
transmon qubit design [24], which has been widely used by both academic and
industry leaders to realize quantum computers. This ratio of $E_{J}/E_{C}$
creates an exponential cutoff for charge fluctuations, leading to longer
lifetimes. Since $E_{J}$ is large compared to $E_{C}$, the quantum Eigen
states are determined in the $\Delta\phi$ basis as shown in Fig. 1(c). The
Eigen energies have an approximate $\sqrt{8E_{J}E_{C}}$ separation, while
first order corrections on the scale of $E_{C}$ create the essential
anharmonicity between energy levels that is required for two-state control
[25]. Therefore the transmon’s $E_{J}/E_{C}$ ratio is large to reduce charge
noise but small enough to prevent excitation beyond the first excited state.

###  II.1 Qubit initialization & readout

The qubit is a quantum mechanical two-level system with logical states
$\ket{0}$ and $\ket{1}$, in analogy to a classical bit. Without any external
or thermal excitation, the superconducting qubit state $\ket{\psi}$ relaxes
into the $\ket{0}$ state. Under a resonant drive $\ket{\psi}$ will oscillate
between $\ket{0}$ and $\ket{1}$ as a superposition on the surface of the Bloch
sphere, shown in Fig. 1(c). The measured period of these so-called Rabi
oscillations is used to calibrate the applied microwave drives for qubit
control. Reading the state of the qubit requires projecting $\ket{\psi}$ onto
the quantization axis. Information about the probability distribution along
other directions is obtained by fast rotations of the axis in question onto
the quantization axis and subsequent measurement. Fast, in this context, means
that the pulse length is short compared to the respective decoherence times.
By measuring in quick succession, the qubit state can be inferred from the
probabilities of the measurement results.

Superconducting qubit states are usually determined using dispersive readout
[26], where $\ket{\psi}$ is not measured directly but is inferred from
measurements of a coupled photon resonator. The interaction between the
resonator and qubit shifts the effective frequency of the resonator by an
amount dependent on the projection of $\ket{\psi}$. Therefore, the qubit state
can be inferred by measurements of the resonator frequency. However the
resonator frequency, $\omega_{r}$, must be detuned from the qubit frequency,
$\omega_{01}$, to prevent measurements from interfering with the qubit state.
The detuning frequency, $\Delta=\omega_{01}-\omega_{r}$, is greater than the
coupling rate, $g$, between the qubit and resonator to ensure that energy is
not coherently exchanged between the qubit and resonator. This condition
prevents a measurement from affecting subsequent measurements (quantum non-
demolition). This control scheme does have a drawback: since the qubit is
coupled to the resonator, noise within the resonator can cause arbitrary phase
decoherence in the qubit. Therefore, measurement signals used to probe the
resonator frequency must be attenuated and filtered to reduce noise and ensure
qubit fidelity.

###  II.2 Qubit manipulation

The microwave tones used for qubit manipulation are referred to as _gates_ or
_pulses_. Qubit manipulation is achieved with a heterodyning technique, where
the pulse signal is generated by a mixer, modulating a baseband signal of a
local oscillator operating close to the desired frequency, with an envelope at
lower frequency. The envelope is generated by fast digital-to-analog
converters, which generate both components of the manipulation or readout
pulses in the respective baseband. For readout the returning microwave signal
from the readout resonator gets down-converted with the same local oscillator
used for the up-conversion, yielding the demodulated baseband signal. After
low pass filtering to suppress leakage of the carrier frequency and further
amplification, the signal is digitized by an analog-to-digital converter card.
Fourier transformation of the incoming signal for both quadratures gives the
complex scattering parameter and in the case of dispersive readout, the state
of the qubit.

Single qubit gates correspond to rotations of a Bloch vector about some axis
of the Bloch sphere while multi-qubit gates take two or more qubits as input
to manipulate at least one qubit state. Multi-qubit gates require entangling
two or more qubits together, while a series of pulses is applied to one or
multiple qubits [29]. Since the qubits are entangled, the readout of one qubit
can be manipulated via a second, ancillary qubit. An example is the Controlled
NOT gate (or CNOT gate), where a target qubit state is flipped if and only if
a second, control qubit state is $\ket{1}$. Quantum logic gates are the
fundamental basic quantum circuit building block, operating on a small number
of qubits (usually one or two). However, it should be noted that due to the
planar structure of superconducting circuitry, connectivity between qubits is
currently limited to nearest-neighbor interactions. This imposes constraints
on gating capabilities, as operations between non-neighboring qubits cannot be
performed. There are current aims to realize 3D-integrated superconducting
circuits [30, 31], which will allow additional connections for beyond-nearest-
neighbor interactions, overcoming the current limitations.

###  II.3 Operational conditions & performance indicators

Like other quantum systems, calculations are limited by the longitudinal and
transverse relaxation times, $T_{1}$ and $T_{2}$. With current technology,
decoherence rates below 1 MHz can be achieved [26], allowing for the creation
and manipulation of single or multiple quantum excitations in superconducting
qubits with fast (nanosecond) control. Improvements to qubit lifetimes have
been achieved primarily through qubit design, improvements in fabrication
quality, and material selection. For current systems, lifetimes are long
enough to ensure computational fidelity. Indeed, for the very best SC QPUs in
the cloud, 1- and 2-qubit gate fidelities exceed 99% with qubit readout errors
in the range of 2-3%.

Regardless of design, qubits must be operated well under the superconducting
transition temperature $T_{c}$. Furthermore, since SC qubits are strongly
coupled to their environment and readout circuitry, thermal and
electromagnetic noise should be reduced as much as possible. Therefore, qubits
are usually measured and operated at $T=10$ mK in dilution refrigerators with
magnetically shielded environments. As mentioned previously, measurement lines
are also typically thermalized and attenuated to reduce noise. The need for
cryogenic environments currently imposes a limitation on the size of SC
quantum computers, since each measurement line leaks heat into to the system
and decreases the effective temperature of the refrigerator. This limitation
can be overcome by implementation of cryogenic processing and multiplexing of
classical signals. Two potential platform solutions are the cryogenic
complementary metal-oxide-semiconducting (Cryo-CMOS) [32, 33, 34] and rapid
single flux quantum (RSFQ) [35] hardware platforms, which can serve as low-
temperature interfaces between classical and quantum systems.

Another limit is the speed at which qubits can be operated. At high
frequencies, superconductivity breaks down as single electrons are excited out
of the superfluid [22]. The presence of these quasiparticles leads to
dissipation and decoherence, and thus qubits are typically designed to operate
at frequencies $\omega_{01}\ll k_{B}T_{c}/\hbar$. For aluminium with $T_{c}=1$
K, qubits are typically designed to operate at $\omega_{01}<20$ GHz.
Additionally, while the macroscopic nature of superconducting qubits allows
for customization of qubit parameters, this benefit comes with a drawback in
producing identical qubits, as small deviations in fabrication uniformity can
be difficult to control.

###  II.4 Use case

Superconducting quantum circuits have been used to simulate many physical
systems. Spin systems have been a particular focus for quantum simulation
through both analog [36, 37, 38, 39] and digital [40, 41] methods. However,
with regard to digital simulations, a recent study [41] performed on an IBM
QPU has concluded that the current state of SC quantum computers is too error-
limited to produce dependable quantitative results for larger (six spins or
more) systems.

Chemical binding energies of molecules have been calculated using VQEs [5, 7,
42, 43] implemented on SC circuits. The VQE method has had relatively good
success in determining binding energies of H2, LiH, BeH2, NaH, KH, and RbH.
Due to the circuitry scale, these studies consider only a limited number of
basis states (e.g. spin orbitals), allowing for a comparison to the exact,
diagonalized solutions. In this context, the calculated results are in good
agreement with theoretical expectations. More recently, binding energies of
hydrogen chains up to H12 have been modeled using Google’s Sycamore QPU [6].
However, it should be noted that several postprocessing techniques were
required to mitigate errors in the raw results and achieve quantitative
chemical accuracy for bonding energies. This work also simulated diazene
(H2N2) isomerization energies for converting cis-diazene to trans-diazene,
marking the first time a chemical transition has been modeled on a quantum
computer. Therefore, despite the limitations from noise and basis size,
digital simulations on SC QPUs show promise for chemical simulations.

##  III Quantum computing with trapped ions

Trapped ions [44, 45] were one of the first platforms proposed for building a
quantum computer as they form a natural representation of an ideal qubit: all
ions are identical by nature, their high degree of isolation from the
environment leads to excellent coherence times and interaction with radio-
frequency (rf) and laser light allows for high-fidelity gate operations.
Qubits are encoded in the electronic states of individual ions trapped by
electric fields in an rf Paul trap. Two-dimensional traps can be micro-
fabricated on silicon chips, called surface traps, and can contain multiple
trapping and interaction zones as well as integrated microwave and laser
access [46, 47, 48, 49]. Interaction of the electronic states of neighbouring
ions is negligibly small [50], but ions are strongly coupled via their motion
which can be exploited to create entanglement between different ions necessary
for multi-qubit gates [51]. Ions are confined in long chains, within which all
ions can interact with each other. Chains can be split and merged, and ions
can be moved across the chip between different zones, providing large
flexibility of connections [52, 53, 54]. Many different elements are used as
ion species, but all ions are typically singly-charged and have a single
remaining valence electron. Popular choices of ion are $\mathrm{Yb}^{+}$,
$\mathrm{Ca}^{+}$ and $\mathrm{Be}^{+}$ [55, 56, 57, 58, 59]. Qubit states can
either both be encoded in ground-state levels (hyperfine- and Zeeman-qubits
[60, 61]) with transition frequencies in the rf range, or with the excited
state encoded in a meta-stable state (eg. $\mathrm{D}_{5/2}$) leading to
optical transition frequencies [62]. Different properties of the atomic
species affect the qubit performance. For example, some hyperfine qubits are
robust to magnetic field noise, which is the main source of decoherence in
trapped ion qubits, and therefore have greatly enhanced coherence times [60,
63]. Other important factors are the existence of low-lying D manifolds, which
can assist readout but cause errors due to scattering in laser gate
operations; the ions’ mass where lighter ions allow faster gates; excited
state lifetimes for optical qubits; and transition frequencies depending on
the availability of suitable lasers.
Scaling up devices from tens to thousands or millions of qubits is arguably
the biggest challenge in realising a quantum computer. The trapped ion
community pursues several paths towards scalability. In the quantum charge-
coupled-device (QCCD) architecture [64, 44] ion chains are broken up into
smaller groups in individual zones, instead of forming a single long string.
For scalability beyond a single chip proposals include connecting separate
traps via photonic links [65, 66, 67] and shuttling of ions across arrays of
chips [68]. Another important ingredient for scalability is the simultaneous
use of different ion species, which allows sympathetic cooling of ions without
affecting the electronic state of the logic and memory qubits [69] and better
spectral isolation for ion-photon entanglement. Strings of ions can be split,
merged and shuttled between different zones with negligible effect on the spin
state and coherence, but a slight increase in ion temperature [52, 53, 54].
While ion traps can be operated at room temperature, their performance is
enhanced at cryogenic temperatures due to a reduction in heating rate and an
increase in ion lifetime. Cooling down to $\sim 10\mathrm{K}$ with liquid
helium cryostats suffices for this purpose.
Trapped ions have the longest coherence times of all contending platforms for
building a quantum computer. Even though their individual operations are
slower than in solid state systems, they still possess a superior ratio of
gate operation time to coherence time, which ultimately results in record
single- and two-qubit gate fidelity. While technology and infrastructure for
solid-state systems is more mature than laser technology due to developments
made for classical computer chips, rapid progress in the stability,
miniaturisation, and integration of laser and ion trap systems has been
achieved in the last few years due to the influx of resources and increase in
demand. Trapped ion quantum computers also benefit from the absence of noisy
direct environments which are present in solid state systems, and the high
degree of connectivity and flexibility of connections in trapped ion systems.
Remaining challenges are to reduce gate errors for larger numbers of qubits,
which tend to increase with the number of ions, and to improve automatisation,
robustness and crosstalk for building larger devices. Further research is also
required in trap fabrication, as one of the major gate error sources stems
from anomalous heating of the ion crystals, thought to be caused by surface
effects on the ion trap electrodes [70, 71, 72, 73].

###  III.1 Qubit readout, initialisation and cooling

Qubits are read out via state-dependent fluorescence detection. All ion
species used for quantum computing have a short-lived excited state that
predominantly decays back into the qubit ground state manifold. For optical
qubits and some hyperfine qubits the qubit frequency is sufficiently large
that the fluorescence laser only couples to one of the qubit states, the
‘bright’ state. Together with selection rules preventing decay from the
excited state into the opposite ‘dark’ qubit state, this allows direct
fluorescence readout. For qubits without direct state selectivity of the
fluorescence laser, the dark state is transferred into a ‘shelf’ state that
does not couple to the fluorescence laser and the excited state. Ion-position
resolved fluorescence can be detected with arrays of photomultiplier tubes or
avalanche photodiodes, on an electron-multiplying charge coupled device camera
[74], or with superconducting nanowire single-photon detectors integrated into
the trap chip [75]. Fluorescence can be collected over a fixed time-bin and
analysed with threshold or maximum likelihood algorithms, or with real-time
analysis and adaptive readout duration. With sufficiently low background
counts and high photon collection and detection efficiency, real-time analysis
achieves the same fidelities as fixed-time threshold analysis, but is
considerably faster [76, 75].

Qubit initialisation is performed via optical pumping, using the same excited
states as for fluorescence readout. Either frequency or polarisation
selectivity is used to ensure that population is excited out of all ground
states apart from the target initial state. Different states can be prepared
by applying a sequence of single qubit operations after optical pumping.

For optimum gate fidelities ion crystals need to be cooled close to their
motional ground states, which is performed with laser cooling. Typically ions
are continuously Doppler cooled during idle time. Before an experiment
resolved-sideband cooling (RSBC) is used to further cool relevant motional
modes to an average motional mode occupation of $\bar{\mathrm{n}}\lesssim
0.1$. Alternatively electromagnetically-induced transparency cooling can be
used to cool all modes simultaneously [77]. While considerably faster than
RSBC, especially for larger ion strings, the final temperature reached is
slightly higher.

###  III.2 Qubit manipulation

Single qubit gates can be driven directly using rf in Zeeman- and hyperfine-
qubits, or using a narrow-linewidth laser to drive the quadrupole transition
in optical qubits. Alternatively a pair of lasers which are far detuned from
the excited state and have the qubit frequency as their frequency difference
can be used to drive qubit rotations via two-photon Raman transitions.
Rotations around the z-axis can be performed trivially by propagating the
phase of all future operations. The phase is defined by a direct digital
synthesis frequency source that is either applied directly on the ions as rf
or controls the frequency, amplitude and phase of the laser beams via an
acousto-optic modulator (AOM). Rf operations couple only very weakly to the
motion due to their low photon energy and can already be performed at Doppler-
cooled temperatures at very high fidelities [60]. They also have superior
phase stability compared to lasers and can easily be integrated into surface
traps, but are harder to address onto single ions.

Multi-qubit gates create entanglement between different qubits and require the
ions’ motion as a bus of interaction between the ions. There are different
schemes for entangling gates, with the most established ones being the closely
related Mølmer-Sørensen (MS) gate [78] and the $\sigma_{z}$ geometric phase
(ZGP) gate[79]. Both create a spin-dependent force on the ions; the MS gate in
the $\ket{+}$ ,$\ket{-}$ basis and the ZGP gate in the $\ket{\uparrow}$ ,
$\ket{\downarrow}$ basis. This force leads to motional excitation and
displacement for one spin parity combination (eg $\ket{\uparrow\downarrow}$ )
but not the other (eg $\ket{\uparrow\uparrow}$ ). Displaced spin states
acquire a phase which ultimately leads to the entanglement. The propagator of
a two-qubit gate with these schemes is diag(1,i,i,1), which corresponds to a
controlled-PHASE gate. This gate can be transformed into a CNOT gate via
additional single-qubit operations. Both gate mechanisms are first-order
insensitive to the ion temperature, which makes them more robust and is an
important factor in the high fidelities achieved. ZGP gates cannot be
performed directly on the low-decoherence clock qubits, but are insensitive to
the absolute magnetic field offset. Two-qubit gates have been performed both
with lasers [62, 57, 59, 80] and rf [81, 82, 83, 84, 85] as well as between
ions of different elements [61, 86, 87, 88]. Due to the weak motional coupling
rf multi-qubit gates are considerably slower than laser gates. Gates can be
performed globally on all ions in a string simultaneously or addressed locally
to a specific subset of ions [58, 55].

###  III.3 Performance indicators

Coherence times in trapped ions are $T_{2}^{*}=50\,\mathrm{s}$ on magnetic-
field insensitive clock qubits [60] and reach over an hour by employing
dynamical decoupling sequences [63]. State-preparation and measurement errors
are $\varepsilon<1\cdot 10^{-3}$, with a mean duration of $46\,\mathrm{\mu s}$
[75, 76, 74], where $\varepsilon\equiv 1-\frac{\mathcal{F}}{100}$ for fidelity
$\mathcal{F}$. Depending on the method and target fidelity, readout times in
these systems may vary between tens and hundreds of $\,\mathrm{\mu s}$. Single
qubit gates have been performed with errors $\varepsilon=1.0(3)\cdot 10^{-6}$
for an rf $\pi/2$-pulse of $12\,\mathrm{\mu s}$ pulse duration on a single ion
[60]. Fast single qubit gates can be implemented with a pulsed laser trading
off fidelity against speed, achieving $\varepsilon=7\cdot 10^{-3}$ for
$t_{\pi/2}=40\,\mathrm{ps}$ [89]. Two qubit gate errors are
$\varepsilon=8(4)\cdot 10^{-4}$ at a gate time of $t_{g}=30\,\mathrm{\mu s}$
[59, 57, 56] for laser gates, and $\varepsilon$ in the interval $[1.7\cdot
10^{-3},0]$ at a gate time of $t_{g}=740\,\mathrm{\mu s}$ for microwave gates
[85]. The fastest two-qubit gates achieved $\varepsilon=2.2(3)\cdot 10^{-3}$
in $t_{g}=1.6\,\mathrm{\mu s}$ [90].

###  III.4 Use case

Various algorithms have been implemented on trapped ion systems, including
Shor’s algorithm and Grover’s search algorithm [91, 92], demonstrations of
error correction [93, 94], analogue quantum simulations, such as the
simulation of many-body dynamical phase transitions [95] exceeding the
capabilities of classical computers, as well as several VQE demonstrations
[96, 97, 98], for example estimating the ground state energy of
$\mathrm{H_{2}}$, LiH and $\mathrm{H_{2}O}$.

Figure 2: IonQ quantum computer based on a chain of trapped ions: A high-
numerical aperture lens allows both individual addressing and readout of the
ions. A multi-channel AOM is used to modulate the amplitude, frequency and
phase of the individual laser beams. Inset: The qubits are encoded in the
hyperfine ground-states
$\ket{\uparrow}\,={{}^{2}\mathrm{S}_{1/2}},\mathrm{F}=1$ and
$\ket{\downarrow}\,={{}^{2}\mathrm{S}_{1/2}},\mathrm{F}=0$ of the trapped
${}^{171}\mathrm{Yb}^{+}\,$ions. Gate operations are performed via a two-
photon Raman process, coupling to the excited ${}^{2}\mathrm{P}$ states
(purple and orange beams). Figure adapted from [97].

Fig. 2 shows the ion trap quantum computer of IonQ, which is commercially
accessible and was used to perform VQE on three individually addressable
qubits encoded in a string of ${}^{171}\mathrm{Yb}^{+}\,$ions to estimate the
ground state energy of the water molecule [97]. The quantum circuit
implementation for the energy-evaluation was optimised to take advantage of
the asymmetric state measurement fidelities of the $\ket{\uparrow}$ and
$\ket{\downarrow}$ states, and the higher fidelity
($\varepsilon_{\phi=\pi/100}\lesssim 4\cdot 10^{-3}$) of partially entangling
gates $\mathrm{XX(\phi)}$ ($\phi<\pi/2$) compared to full entangling gates
$\mathrm{XX(\pi/2)}$ ($\varepsilon\lesssim 4\cdot 10^{-2}$). The longest
implemented circuit comprised 6 CNOT two-qubit operations. An energy
uncertainty close to the chemical uncertainty of $1.6\,\mathrm{mHa}$ was
achieved (albeit in a minimal basis set), without a need for implementing
error mitigation techniques such as Richardson extrapolation [99].

##  IV Silicon quantum computer

Today’s digital age is enabled by the relentless progress and optimisation of
semiconductor materials and technology. From an industrial standpoint, the use
of well-established nanofabrication techniques for the development of quantum
machines would be economically attractive to achieve large-scale systems. As
discussed, some of these manufacturing techniques are already applied to
superconducting and ion trap quantum platforms, and are expected to become
central for the development of silicon-based systems, offering the prospect of
integrating millions of qubits on chips at affordable manufacturing costs,
akin to classical commercial electronics. Besides this technological
motivation, silicon is a particularly suitable material for spin-based quantum
devices from a performance viewpoint. Through isotopic purification, the only
isotope bearing a nuclear spin (29Si) in natural silicon can be nearly
completely removed, making the silicon crystal a quasi-spin-noise-free
environment for the qubit. This results in silicon spin-qubits having the
longest coherence time among solid-state implementations.
Besides silicon, there exists a large variety of semiconductor systems
currently under investigation for quantum computing applications [100, 101,
102, 103]. The main differences lie in the type of material (e.g. natural or
purified silicon, synthetic diamond, silicon carbide, heterostructures such as
GaAs/AlGaAs, Si/SiGe or Ge/SiGe), the operational conditions (ranging from
room temperature down to millikelvin temperature), the way each qubit is
spatially confined within the material (e.g. gate-defined quantum dots, etched
nanowires, atomic-size crystallographic defects, implanted dopant impurities),
the way the qubit state is readout (e.g. electrical readout via charge
sensors, or optical readout through photoluminescence), and the way the qubit
state is manipulated (e.g. electron spin resonance via magnetic field pulsing,
electric dipole spin resonance via electric field pulsing). Despite such
diversity, a common denominator in most platforms is the choice of
electron/hole spins as the two-level system embodying the qubits. The
paradigmatic encoding is represented by a single spin in a static magnetic
field with its two Zeeman-split energy levels representing the states
$\ket{0}$ and $\ket{1}$. Other implementations that have been explored include
two-electron singlet-triplet qubits, three-electron charge-spin hybrid qubits
and three-electron exchange-only qubits. Such rich ecosystem gives rise to
significant performance variations among qubit implementations. The trade-offs
can be many, including the robustness to specific noise sources and the ease
of operation. The coherence times can range from few tens of nanoseconds in
GaAs/AlGaAs quantum dots to few seconds in silicon dopants, and the single-
qubit gate time can vary between sub-nanosecond and hundreds of nanoseconds in
Si/SiGe quantum dots and silicon dopants, respectively.
In this Section, we are going to focus our attention on a particular type of
semiconductor qubit system, which has been deployed for the realisation of the
first spin-based quantum computer in the Cloud: SPIN-2QPU [104], developed at
QuTech (a collaboration between TUDelft and TNO). It consists of two single
electron spin qubits in a double quantum dot (DQD) that is electrostatically
defined by metallic gate electrodes deposited on top of an isotopically
purified Si/SiGe heterostructure, as illustrated in Fig. 3 (a) and Fig. 3 (b).
Similar to the other quantum processors discussed previously, spin-based
machines must meet certain functional criteria. These include reliable
initialisation to a known state, high fidelity projective readout of the final
state, and qubit manipulation through high-quality single- and two-qubit
gates. Let us see how SPIN-2QPU satisfies these criteria.

###  IV.1 Qubit initialisation & readout

The readout of the qubit state is ultimately a measurement of the electron
spin orientation. However, the magnetic moment of a single spin is exceedingly
small and its direct detection quite difficult. By contrast, the detection of
small displacements of single charges is routinely carried out in
semiconductor devices. To this end, SPIN-2QPU uses a single-electron
transistor (SET) capacitively coupled to the DQD, as shown in Fig. 3 (b).
Whenever a single electron leaves/enters the DQD, the SET produces a discrete
jump in the value of its electric current caused by a change in its operation
point.

Figure 3: (a) Schematic cross-sectional view of a DQD device used to control
two spin qubits. Top metal gates (yellow areas) are used to tune the
conduction band profile (dashed line) in the Si layer and isolate two electron
spins. An n-type doped region of semiconductor (pink shaded area) is used as
an electron reservoir tunnel-coupled to the left QD. A layer of cobalt (blue
box) is deposited on top of the gate layers to generate a controlled magnetic
field gradient across the DQD. (b) SEM micrograph of the DQD in (a). The
aluminum gates are patterned with electron-beam lithography. The two qubits
are formed under the gates highlighted with red and blue circles. The SET
detector is formed under the gates highlighted in yellow. Gates that
accumulate the electron reservoirs for the DQD and the SET are connected to
Ohmic contacts and highlighted by crossed squares. Dashed lines indicate the
region where the micromagnet is deposited. (c) Schematic diagram of energy
levels for the left QD and the electron reservoir during the readout pulse
sequence. Energy levels in the QD are Zeeman-split according to spin
polarization. (d) Pulsing sequence (top) for the spin readout and normalised
SET signal for spin-up (middle) and spin-down (bottom) qubit states. Panels
(a) and (b) are adapted from Ref. [104]. Panels (c) and (d) are adapted from
Ref. [105].

Reading out the spin state is, therefore, a matter of making a so-called spin-
to-charge conversion, whereby the electron is allowed to tunnel in or out the
DQD in a way that depends on its spin state, equivalent to whether the qubit
is in state $\ket{0}$ or $\ket{1}$ . As shown in Fig. 3 (c), the selection
rule is energy-based. A single spin in one of the dots is capacitively coupled
to the SET and tunnel coupled to a reservoir. After spin manipulation, the
dot’s energy level is tuned with a gate voltage pulse such that the Fermi
reservoir lies between the two Zeeman-split spin states. If the electron is in
state $\ket{\downarrow}$, it does not have enough energy to leave the dot, and
there is no SET current change due to a lack of charge rearrangement. For a
state $\ket{\uparrow}$, the electron can tunnel out of the quantum dot and
into the reservoir, leading to a change in SET current until a new electron
tunnels in and re-initializes the qubit to its ground state. Current traces
for these two alternative scenarios are shown in Fig. 3 (d). Note that in this
system initialization can be seen as a by-product of readout, given that an
electron with a known spin, i.e. $\ket{\downarrow}$, always resides in the dot
at the end of the sequence.

###  IV.2 Qubit manipulation

Analogously to other qubit realisations, a spin-qubit requires independent
rotations about the axes of the Bloch sphere (single-qubit gate), as well as
rotations that are dependent on the state of another qubit (two-qubit gate),
in order to form a set of universal quantum gates. Through a two-qubit gate,
entangled states can be created when one of the two qubits starts in a
superposition of states.
SPIN-2QPU carries out single-qubit gate operations through electric dipole
spin resonance (EDSR). It consists of a microwave modulated electric pulse
delivered through a gate electrode that oscillates the electron wavefunction.
This has the effect of rotating the electron spin whenever the electron
experiences a time-varying magnetic field resonant with its Zeeman splitting.
This requires the presence of a synthetic spin-orbit field obtained through a
local magnetic field gradient in the DQD, which is engineered by depositing a
cobalt micromagnet on top of the device gate layer (see Fig. 3 (a)). The
amplitude of the EDSR pulse controls the spin vector’s rotation frequency
around the Bloch sphere, its phase controls the rotation axis, and its
duration controls the rotation angle. The frequency of the pulse allows one to
select which qubit is manipulated, given that each electron experiences a
slightly different magnetic field due to the different position within the
DQD.
SPIN-2QPU carries out two-qubit gate control via modulation of the exchange
interaction. The idea is to quickly turn on the tunnel coupling between two
neighbouring spins by applying a gate-voltage pulse that lowers the tunnel
barrier between their corresponding quantum dots, so that the electron
wavefunctions overlap. Such overlap leads to an exchange interaction between
the spins, which can be exploited for conditional gate operations.

###  IV.3 Operational conditions & performance indicators

The readout protocol is effective if the qubit energy levels are separated by
at least a few times the thermal energy. This is ultimately the reason why
SPIN-2QPU and similar semiconductor-based quantum systems need to be operated
at dilution refrigerator temperature ($T$) and in the presence of an external
static magnetic field ($B$). Typical conditions require $B\approx$ 1 T and
$T\approx$ 50 mK. The duration of the readout sequence is ultimately
determined by the tunnelling rate between the DQD and the reservoir, as well
as by the bandwidth of the SET detector. SPIN-2QPU’s readout duration is
$\approx$ 300 $\mu$s per qubit and its readout fidelity is approximately 85%.
Given a single-qubit gate duration of approximately $250$ ns and a phase
coherence time of at least 6 $\mu$s, SPIN-2QPU achieves single-qubit
fidelities in excess of 99.0%. As for 2-qubit operations, the only allowed
native gate is CZ. Hence, other gates like CNOT and SWAP have to be decomposed
into CZ operations in combination with single-qubit rotations. This comes at
the expense of fidelity and operational time. A detailed benchmark for CZ is
ongoing. Preliminary data show gate duration of around 150 ns and fidelity in
excess of 90%, but this latter figure is likely to be a conservative
underestimate at this stage.

###  IV.4 Use case

At present, we are not aware of VQE simulations carried out with SPIN-2QPU or
any other semiconductor qubit system, possibly due to the limited qubit count.
By contrast, 2D arrays of semiconductor QDs have been used for analog
simulations of magnetic and insulating materials by spatially engineering
Hamiltonians onto the array [106, 107]. It is, however, useful to report that
it has been possible to run digital algorithms of different kinds
(Deutsch–Josza and Grover) on the SiGe processor that QuTech used to prototype
SPIN2-QPU [108]. This ultimately casts a positive light for future uses of
semiconductor machines in computational chemistry.

Manufacturer |  Platform |  Cloud access |  Max # qubits |  Gate fidelity (1-qubit, 2-qubit) |  Max QV |  Simulated molecules
---|---|---|---|---|---|---
IBM |  Superconducting |  IBM Quantum Experience (Open access) |  15 (Melbourne) |  99.97%, 99.16% (Santiago) |  32 (Santiago) |  H2, LiH, BeH2, NaH, KH, RbH
IonQ |  Trapped Ions |  Microsoft Azure or Amazon Bracket |  11 |  99.50%, 97.50% |  not published |  H2O
QuTech |  Silicon |  Quantum Inspire |  2 (Spin2-QPU) |  $\approx$ 99%, $\approx$ 90% |  not published |  none
Google |  Superconducting |  Google Quantum AI |  53 (Sycamore) |  99.85%, 99.35% |  not published |  H2N2, H6, H8, H10, H12
Rigetti |  Superconducting |  Rigetti Quantum Cloud |  31 (Aspen-8) |  99.8%, 95.9% |  8 (Aspen-4) |  NaH, H2
Honeywell |  Trapped Ions |  Microsoft Azure or Amazon Bracket |  10 (H1) |  99.97%, 99.5% |  128 |  none
Table 1: Quantum computing hardware in the cloud (a non-exhaustive selection).
Wherever more than one QPU is available, the relevant machine is indicated
within brackets. Simulated molecules column denotes experiments run with any
quantum machine from the relevant manufacturer, not necessarily one of those
listed. IBM hardware considered is limited to Open Access services. Google
cloud services are limited to emulators at present, although the reported
chemical simulations have been performed with proprietary physical hardware.

##  V Discussion

In recent months significant attention has been drawn to superconducting
quantum hardware because a team at Google achieved a much anticipated
milestone, namely quantum supremacy [109]. By quantum supremacy, it is meant
that a quantum computer is able to produce the solution to a computational
problem that would be otherwise impossible in a reasonable time with a
classical machine. Google scientists achieved this with a 53-transmon-qubit
processor (Table 1) by showing efficient sampling of random quantum circuits.
While this result is of primary importance for the field as a whole, the
problem tackled did not bear any relevance to molecular chemistry. Therefore,
with regard to this type of problem, a quantum advantage is yet to be
demonstrated. However, in a more recent study [6], the Google team used the
same quantum processor for chemical simulations, as discussed in Section II.4.
They demonstrated the most complex ground state simulation to date with as
many as 24 spin-orbitals mapped onto 12 qubits. Although these calculations
are relatively straightforward with a conventional supercomputer, they
represent a significant advance of the state-of-the-art in quantum computing
power, as the number of qubits used and orbitals simulated in prior
experiments was no more than six [7]. While Google’s quantum hardware is
scheduled to be deployed onto cloud services imminently, there is already a
variety of tools made available by Google scientists to experiment with
emulated hardware tailored for applications in molecular chemistry[110]. As
for superconducting hardware readily available in the cloud, one has to
currently turn to IBM or Rigetti, see Table 1. IBM has about a dozen QPUs in
the cloud, arguably the most extensive offer yet. Just through its Open Access
service, the community can access eight machines with qubit counts ranging
from 1 to 15 and QV ranging from 8 to 32. The most powerful QPUs with qubit
counts up to 65 and QV up to 32 are available for business clients via Premium
Access. A recent breakthrough has led to QV=64 for a new 27-qubit system not
yet available in the cloud. IBM scientists were among the pioneers in
exploiting QPUs for molecular chemistry applications (see Table 1) [7]. More
recently, they have also shown that improved simulation accuracy can be
obtained by adopting error mitigation techniques at algorithmic level [8].
This is important because it can be used to enhance the computational power of
a processor without any hardware modification.
Quantum machines based on trapped ions have progressed very quickly in the
past year alone. While devices used for digital quantum computing typically
have a lower qubit count than their superconducting counterparts, analogue
quantum simulation has been performed on strings containing up to 53 qubits
[95] and single qubit operations have been performed in devices containing up
to 79 qubits [111]. Due to superior gate fidelity and qubit-to-qubit
connectivity, the quantum volume of ion trap processors is outperforming
superconducting devices even for smaller numbers of qubits. Recently,
corporate research teams at IonQ and Honeywell have made QPUs available
through the wider cloud services of Amazon and Microsoft, see Table 1.
Honeywell’s QPU shows the largest volume to date, i.e. QV=128. Both Honeywell
and IonQ have recently announced the imminent launch of upgraded QPUs with
significantly enhanced QV values.Trapped ion machines have also been used for
molecular chemistry simulations[96, 97, 112]. The most complex molecular
simulation performed to date with trapped ions is the evaluation of the
binding energy of the water molecule with a 3-qubit QPU from IonQ [97], as
discussed in Section III.
The 2-qubit silicon quantum processor made by QuTech is the only spin-based
system in the cloud. The service through which it is accessible, the platform
Quantum Inspire, also provides a more powerful alternative based on a 5-qubit
superconducting QPU. Silicon SPIN2-QPU has been the latest to be deployed
(April 2020) and is not yet fully characterised, hence only approximate
fidelities are quoted in Table 1. Although no chemical simulations have been
attempted yet, one should expect that the semiconductor community will soon
fill this gap. The modest qubit count should not be an insurmountable
impediment if one considers that early 2-qubit QPUs were successfully used to
simulate diatomic molecules [5, 3, 4]. Undoubtedly, Si-based machines have yet
to cover much ground before becoming realistic competitors of the other two
major platforms. For example, high-fidelity single- and two-qubit gates have
only recently been achieved and are not yet on par with those of the other
hardware platforms [101]. Furthermore, qubit variability due to atomic level
defects in the material and its interfaces is an issue that currently hampers
scalability. Nonetheless, the interest around these devices is justified by
the fact that in principle they can be manufactured with industrial CMOS
technology, and have the smallest qubit footprint [113]. This bodes well for
future upgrades of such systems towards the million-qubit-machines needed for
useful applications. Finally, note that there exists another type of silicon
QPU based on photonic technology (as opposed to spins) with two systems
accessible via cloud services [114, 115].

##  VI Conclusion and Outlook

A lot of theoretical and experimental ground has been covered since the early
80s, when Feynman proposed to use controllable quantum devices for
computational problems in chemistry and physics. There are now dozens of
small-scale quantum computers in the cloud and many more in academic and
corporate laboratories worldwide. The electronic structures of simple
molecules ranging from diatomic systems to chains of a dozen atoms has been
determined with several QPU incarnations.
In this Article, we have discussed the hardware of the most popular types of
quantum computers, for which we have summarised the main techniques for
physical encoding, manipulation and readout of quantum information. We have
paid particular attention to the machines that the reader could easily access
via cloud services, i.e. superconducting-, trapped ion- and silicon-based
processors. For these, we have described the main performance specifications
and operational conditions. Our target has been to highlight to what extent
these early prototypes have been employed for chemistry simulations. The
underlying message is that, despite relentless progress, none of the machines
built thus far is yet advantageous to a chemist, if compared to classical
computational methods. What needs to happen to change this?
In order to achieve a sizable quantum advantage in computational chemistry
with NISQ machines, the coordinated efforts between quantum hardware and
quantum algorithm developers will need to continue if not intensify. Hardware
improvements in terms of qubit count, qubit connectivity, quantum gate speed
and fidelity, as well as overall QPU volume will be a central focus for years
to come. These advances will be essential to bring quantum simulation run-
times down to practical length-scales [2]. However, recent breakthroughs [8,
6] have also shown that tailoring algorithms to the specific quantum hardware
available in combination with error mitigation techniques could be important
for accurate chemical computation on near-term machines. Particularly,
restrictions to realisable gates inherent to NISQ processors could be bypassed
with ad-hoc compilation methods.
Beyond the NISQ era, i.e. without today’s limitations due to noise, there will
be the possibility of taking full advantage of the computational speed-up of
quantum systems. QEC protocols will have to be reliably implemented to produce
such step change. During this transition, a risk to be avoided will be that
today’s capability restrictions, rather than being lifted altogether, will be
merely transferred from the quantum layer onto the classical control layer
[116]. There are two complementary considerations to this potential problem.
Firstly, QEC will require fast feedback between measurement and control, and
communication latency may become an issue. If there is a sizable physical
distance between the quantum hardware and the classical control hardware,
which is likely for cryogenic QPUs, delays in the communication lines may pose
a synchronization challenge if they become of the same order as the gate time.
Secondly, a computational bottleneck may occur in handling error correction
cycles for large number of physical qubits. For example, a QPU with a million
qubits corrected with cycles of 1 $\mu$s will require classical information
processing at a bandwidth of 1 Tbit/s. If both latency and bandwidth issues
are to be solved at once, novel ultra-low-power cryogenic RAM and CPU may need
to be developed, so that they could sit near or within the same chip of a
cryogenic QPU without generating detrimental heat loads [113].
We believe that the challenges described do not represent a fundamental
roadblock towards large-scale fault-tolerant quantum computing. However, they
do pose significant engineering hurdles that will require synergies between
quantum and electronic engineers, as well as quantum software developers and
end users. We hope that this Article will trigger the curiosity of theoretical
and quantum chemists in trying out the available cloud machines, get involved
into the ongoing conversation and, eventually, steer quantum systems
development to the benefit of their scientific agenda.

###### Acknowledgements.

We wish to thank N. Samkharadze for useful discussions. AR acknowledges the
support of the UK Government Department for Business, Energy and Industrial
Strategy through the UK National Quantum Technologies Programme, as well as
support from a UKRI Future Leaders Fellowship (MR/T041110/1). VMS acknowledges
funding from Christ Church College, Oxford. PGB and MW acknowledge support
from the EPSRC Hub in Quantum Computing and Simulation (EP/T001062/1).

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