Quantum computing hardware in the cloud: Should a computational chemist care?
QQuantum computing hardware in the cloud:Should a computational chemist care?
Alessandro Rossi,
1, 2, ∗ Paul G. Baity, Vera M. Sch¨afer, and Martin Weides Department of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom National Physical Laboratory, Hampton Road, Teddington TW11 0LW, United Kingdom James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, United Kingdom Department of Physics, University of Oxford, Clarendon Laboratory,Parks Road, Oxford OX1 3PU, United Kingdom (Dated: February 8, 2021)Within the last decade much progress has been made in the experimental realisation of quantumcomputing hardware based on a variety of physical systems. Rapid progress has been fuelled bythe conviction that sufficiently powerful quantum machines will herald enormous computationaladvantages in many fields, including chemical research. A quantum computer capable of simulatingthe electronic structures of complex molecules would be a game changer for the design of newdrugs and materials. Given the potential implications of this technology, there is a need within thechemistry community to keep abreast with the latest developments as well as becoming involved inexperimentation with quantum prototypes. To facilitate this, here we review the types of quantumcomputing hardware that have been made available to the public through cloud services. We focuson three architectures, namely superconductors, trapped ions and semiconductors. For each onewe summarise the basic physical operations, requirements and performance. We discuss to whatextent each system has been used for molecular chemistry problems and highlight the most pressinghardware issues to be solved for a chemistry-relevant quantum advantage to eventually emerge.
I. INTRODUCTION
This year marks exactly 40 years since Richard Feyn-man famously said [1]: “Nature isn’t classical, dammit,and if you want to make a simulation of nature, you’dbetter make it quantum mechanical, and by golly it’sa wonderful problem, because it doesn’t look so easy”.On the one hand, the visionary physicist anticipated thepossibility (and the inherent difficulty) of building a newtype of computing apparatus operating according to thelaws of quantum mechanics. On the other hand, he hadimmediately identified one of its most useful areas of ap-plication, i.e. simulations of chemical and physical sys-tems.Computational chemists will indeed benefit from fu-ture quantum computers for calculations of molecularenergies to within chemical accuracy, defined to be thetarget accuracy necessary to estimate chemical reactionrates at room temperature ( ≈ ∗ [email protected] and material chemistry has been relentless. Small-scalequantum machines developed by academic or corporateresearch centres have been initially used to simulate sim-ple diatomic or triatomic molecules made up of just Hand He atoms [3–5]. Recently, more powerful quantumcomputers have been used to simulate larger compoundscontaining N, Li and Be atoms [6–8]. Although thesestudies do not show a clear advantage in using quantumcomputing over the conventional computational methodsthat have been used for their validation, they do indicatethat hurdles are being tackled and viable ways forwardare becoming available.The major impediments that currently stifle quantumcomputers are limits to the number of computationalunits and computational errors. The units of quantuminformation are called quantum bits (qubits) in anal-ogy with the binary bits of classical computers. Quan-tum algorithms for chemical calculations use qubit-basedHamiltonians to map molecular many-body Hamiltoni-ans and evaluate the system wavefunction through re-peated sampling of the qubit register states [9–11]. Oneparticular algorithm, namely the variational quantumeigensolver (VQE) [12], has acquired prominence becauseit alleviates the computational burden on today’s lim-ited quantum machines by using a classical co-processorto support the calculation. To date, the most advancedVQE simulations have mapped just 24 molecular orbitalsonto 12 qubits [6], a relatively easy feat for traditionalcomputers. In order to calculate the energy ground stateof more complex systems with chemical accuracy, it isexpected that the number of qubits available will need toincrease by orders of magnitude. A recent estimate [2]indicates that more than 1500 spin-orbitals are requiredfor a VQE calculation that could outperform classical su- a r X i v : . [ qu a n t - ph ] F e b per computers.The other hurdle to consider is that qubits are error-prone due to noise-limited phase coherence. Ultimately,there is a limit to the number and duration of operations(also known as qubit gates) that a quantum computercan carry out before error propagation leads to computa-tional failure. Quantum error correction (QEC) schemesto correct these errors have been identified [13, 14]. Themain drawback is that QEC leads to hardware aggrava-tion, given that several physical qubits are required torealise a single error-corrected “logical” qubit. Some es-timates based on realistic qubit noise levels conclude thatthe ratio of physical to logical qubits to reach fault tol-erant machines could be as high as 1,000 : 1 [15].It is, therefore, evident that, to approach quantumchemistry simulations in a meaningful way, quantumcomputers with millions of physical qubits will be re-quired, if one has to accurately map thousands of spin-orbitals. By contrast, today’s quantum computers relyon a small number of noisy qubits (less than 100 atpresent) because the ability to manufacture, intercon-nect and error-correct qubits on larger scales is not yetsufficiently developed. This is why quantum machinesare presently dubbed NISQ (Noisy Intermediate StageQuantum) [16]. An important figure of merit for NISQsystems is called quantum volume (QV) [17], which com-bines in one convenient metric the number of qubits avail-able, how extensively they are interconnected, and theirgate fidelity. A larger QV indicates that more complexquantum algorithms can be successfully run. This metricclearly shows that, to increase the computational power,is not sufficient to build machines with more qubits ifthese remain affected by high levels of noise. Hence, thechallenge of improving quantum computing power is acoordinated effort in scaling up qubits, making them asinterconnected as possible, and reducing the error rates.NISQ computers come in a variety of hardware imple-mentations. Different from classical computers for whichthe Central Processing Unit (CPU) is invariably madewith silicon integrated technology, Quantum ProcessingUnits (QPU) can also be realised with superconductormicrochips, ions or neutral atoms trapped in a vacuum,and on-chip photonic waveguides. Different technologiespresent 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 quan-tum hardware types. Specifically, we will look at digitalprogrammable QPUs, as opposed to adiabatic or analogsystems [18, 19]. Among these, we shall discuss comput-ers available to the general public through cloud services.On the one hand, being available to the public, and notjust to specialised quantum developers, indicates thesesystems have reached superior maturity. On the otherhand, we feel that a description of how these systems op-erate at hardware level will benefit the reader who mayhave to navigate through offers and subscription packagesto identify the most relevant service for the computa-tional chemistry application of interest. This may indeed become a daunting task without prior knowledge giventhe pace with which these services are becoming availableand compete to acquire large customer bases. Globalcorporations offering cloud access include Google, IBM,Microsoft and Amazon. We shall limit our discussion tothree types of hardware in the cloud: superconductor-,ion trap- and silicon-based quantum computers. For eachone of these systems we discuss how qubits are physi-cally embodied, initialised, read and manipulated. Wewill describe the operational requirements and the mainperformance parameters of each implementation. We willprovide some use cases relevant to quantum chemical sim-ulations to exemplify the usefulness of different machinesin relevant contexts. The remainder of this Article is or-ganised as follows. Superconductor devices are describedin Section II, ion trap systems in Section III, and a siliconprocessor in Section IV. These technologies are comparedin Section V, and finally an outlook for future develop-ments is discussed in Section VI. II. SUPERCONDUCTING QUANTUMCOMPUTERS
Superconducting (SC) circuits are the most widelyused systems for quantum computing. Many industryleaders, such as Google, IBM, and Rigetti, use supercon-ducting quantum circuits to realize their quantum com-puters. Qubits implemented on superconducting devicesfulfil the requirements [20] for scalable quantum com-puting, and therefore micron-sized quantum circuits andassociated integrated-circuit processing techniques canbe scaled up when implemented using superconductingquantum technologies. Whereas trapped ion and silicondevices control and read (sub-)atom scale componentsas their quantum systems, in SC circuits information isencoded into a macroscopic quantum state of the conden-sate of paired electrons (so-called Cooper pairs), whichcollectively participate in a charged superfluid state witha wave function Ψ( (cid:126)r, t ) = | Ψ( (cid:126)r, t ) | e iφ ( (cid:126)r,t ) [21, 22]. Here,the wave function parameters | Ψ( (cid:126)r, t ) | and φ ( (cid:126)r, t ) de-scribe the density of Cooper pairs and complex phaseof the condensate as a function of position (cid:126)r and time t . Superconducting qubits, such as the one shown inFig. 1(a), consist of islands of superconducting material,such as aluminium, connected by one or more Joseph-son junctions [22], which are nm-thin insulating barriersmade from e.g. aluminium oxide. The current I passingthrough the Josephson junction depends on the phasedifference ∆ φ between the superconductors at either sideof the junction by the relation I = I sin(∆ φ ), where I is the largest supercurrent supported by the junction.When a voltage difference V occurs across the junction,∆ φ changes as d ∆ φdt = 2 eV / (cid:126) [22, 27], where e is theelectron charge and (cid:126) is the reduced Planck’s constant.This time dependence leads to non-linear resonance be-havior with quantized states that are determined by flux,charge, and phase degrees of freedom [28]. FIG. 1. (a) Concentric transmon qubit design from Ref. [23]and (inset) its equivalent circuit diagram. Two superconduct-ing islands (green and blue) are shunted by a small Josephsonjunction bridge (orange). The qubit state is read out using acoplanar waveguide resonator (red). This readout resonatoris inductively coupled to a signal line (black). (b) The statesof the transmon qubit are determined by the sinusoidal po-tential (black solid line) of the Josephson junction. Solved inthe phase basis (∆ φ ), the Eigen energies (solid colored lines)can be approximated by a harmonic oscillator (dashed lines,respective colors) whose degeneracies are lifted by first ordercorrections from capacitive charging energy on the junction[24–26]. (c) Diagram of the Bloch sphere. The ground | (cid:105) and first excited | (cid:105) states are used to define the qubit’s logi-cal state | ψ (cid:105) , which is a linear combination of | (cid:105) and | (cid:105) withrespective complex amplitudes α and β . | ψ (cid:105) can be manip-ulated by voltage pulses and gating operations and read outby projection onto a specified measurement basis. The effective circuit diagram of a superconductingqubit is shown in the inset of Fig. 1(a) and can be de-scribed by the Hamiltonian [24, 26] H = 4 E C (∆ n ) − E J cos(∆ φ ) , (1)where E J is the energy of the current passing throughthe junction and E C is the capacitive charging energy be-tween the two superconducting islands. Quantum statesare usually determined in either the basis of ∆ φ , as shownin Fig. 1(b), or capacitive charge number ∆ n , dependingon the relative strengths of E J and E C . Similar to theconjugate variables of position and momentum, ∆ φ and∆ n have a non-zero commutation [∆ φ, ∆ n ] = i [26] anduncertainty relation σ ∆ φ σ ∆ n (cid:38) | (cid:105) and firstexcited | (cid:105) states.One of the benefits of superconducting qubits is theability to engineer a wide range of operational parame-ters by tuning the parameters E J and E C through in-tentional design choices. Perhaps the most widely useddesign choice is to have E J /E C ∼ . This is the so-called transmon qubit design [24], which has been widelyused by both academic and industry leaders to realizequantum computers. This ratio of E J /E C creates an ex-ponential cutoff for charge fluctuations, leading to longerlifetimes. Since E J is large compared to E C , the quantumEigen states are determined in the ∆ φ basis as shownin Fig. 1(c). The Eigen energies have an approximate √ E J E C separation, while first order corrections on thescale of E C create the essential anharmonicity betweenenergy levels that is required for two-state control [25].Therefore the transmon’s E J /E C ratio is large to reducecharge noise but small enough to prevent excitation be-yond the first excited state. A. Qubit initialization & readout
The qubit is a quantum mechanical two-level systemwith logical states | (cid:105) and | (cid:105) , in analogy to a classicalbit. Without any external or thermal excitation, the su-perconducting qubit state | ψ (cid:105) relaxes into the | (cid:105) state.Under a resonant drive | ψ (cid:105) will oscillate between | (cid:105) and | (cid:105) 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 appliedmicrowave drives for qubit control. Reading the stateof the qubit requires projecting | ψ (cid:105) onto the quantiza-tion axis. Information about the probability distributionalong other directions is obtained by fast rotations of theaxis in question onto the quantization axis and subse-quent measurement. Fast, in this context, means thatthe pulse length is short compared to the respective de-coherence times. By measuring in quick succession, thequbit state can be inferred from the probabilities of themeasurement results.Superconducting qubit states are usually determinedusing dispersive readout [26], where | ψ (cid:105) is not measureddirectly but is inferred from measurements of a coupledphoton resonator. The interaction between the resonatorand qubit shifts the effective frequency of the resonatorby an amount dependent on the projection of | ψ (cid:105) . There-fore, the qubit state can be inferred by measurementsof the resonator frequency. However the resonator fre-quency, ω r , must be detuned from the qubit frequency, ω , to prevent measurements from interfering with thequbit state. The detuning frequency, ∆ = ω − ω r ,is greater than the coupling rate, g , between the qubitand resonator to ensure that energy is not coherentlyexchanged between the qubit and resonator. This condi-tion prevents a measurement from affecting subsequentmeasurements (quantum non-demolition). This controlscheme does have a drawback: since the qubit is coupledto the resonator, noise within the resonator can cause ar-bitrary phase decoherence in the qubit. Therefore, mea-surement signals used to measure the resonator frequencymust be attenuated and filtered to reduce noise and en-sure qubit fidelity. B. Qubit manipulation
The microwave tones used for qubit manipulation arereferred to as gates or pulses . Qubit manipulation isachieved with a heterodyning technique, where the pulsesignal is generated by a mixer, modulating a basebandsignal of a local oscillator operating close to the desiredfrequency, with an envelope at lower frequency. The en-velope is generated by fast digital-to-analog converters,which generate both components of the manipulation orreadout pulses in the respective baseband. For read-out the returning microwave signal from the readout res-onator gets down-converted with the same local oscilla-tor used for the up-conversion, yielding the demodulatedbaseband signal. After low pass filtering to suppress leak-age of the carrier frequency and further amplification, thesignal is digitized by an analog-to-digital converter card.Fourier transformation of the incoming signal for bothquadratures gives the complex scattering parameter andin the case of dispersive readout, the state of the qubit.Single qubit gates correspond to rotations of a Blochvector about some axis of the Bloch sphere while multi-qubit gates take two or more qubits as input to manip-ulate -at least- one qubit state. An example is the Con-trolled NOT (or CNOT gate) flipping the second qubit(the target qubit) if and only if the first qubit (the controlqubit) is | (cid:105) . Quantum logic gates are the fundamentalbasic quantum circuit operating on a small number ofqubits (usually one or two). They are forming the basisfor quantum algorithms, which act on the input qubitsand terminate with a measurement. C. Operational conditions & performanceindicators
Like other quantum systems, calculations are limitedby the longitudinal and transverse relaxation times, T and T . With current technology, decoherence rates be-low 1 MHz can be achieved [26], allowing for the cre-ation and manipulation of single or multiple quantumexcitations in superconducting qubits with fast (nanosec-ond) control. Improvements to qubit lifetimes have beenachieved primarily through qubit design, improvementsin fabrication quality, and material selection. For the sys-tems allowing cloud access, 1- and 2-qubit gate fidelitiestypically exceed 99%.Regardless of design, qubits must be operated well un-der the superconducting transition temperature T c . Fur-thermore, since SC qubits are strongly coupled to theirenvironment and readout circuitry, thermal and electro-magnetic noise should be reduced as much as possible.Therefore, qubits are usually measured and operated at T = 10 mK in dilution refrigerators with magneticallyshielded environments. As mentioned previously, mea-surement lines are also typically thermalized and atten-uated to reduce noise. The need for cryogenic environ-ments currently imposes a limitation on the size of SCquantum computers, since each measurement line leaksheat into to the system and decreases the effective tem-perature of the refrigerator. Overcoming this limitationis an essential requirement for scaling up superconduct-ing quantum circuits.Another limit is the speed at which qubits can be op-erated. At high frequencies, superconductivity breaksdown as single electrons are excited out of the super-fluid [22]. The presence of these quasiparticles leads todissipation and decoherence, and thus qubits are typi-cally designed to operate at frequencies ω (cid:28) k B T c / (cid:126) .For aluminium with T c = 1 K, qubits are typically de-signed to operate at ω <
20 GHz. Additionally, whilethe macroscopic nature of superconducting qubits allowsfor customization of qubit parameters, this benefit comeswith a drawback in producing identical qubits, as smalldeviations in fabrication uniformity can be difficult tocontrol.
D. Use case
Superconducting quantum circuits have been used tosimulate many physical systems. Spin systems have beena particular focus for quantum simulation through bothanalog [29–32] and digital [33, 34] methods. However,with regard to digital simulations, a recent study [34]performed on an IBM QPU has concluded that the cur-rent state of SC quantum computers is too error-limitedto produce dependable quantitative results for larger (sixspins or more) systems.Chemical binding energies of molecules have been cal-culated using VQEs [5, 7, 35, 36] implemented on SCcircuits. The VQE method has had good success in de-termining those of H , LiH, BeH , NaH, KH, and RbH.More recently, binding energies of hydrogen chains up toH have been modeled using Google’s Sycamore QPU[6]. However, it should be noted that several postpro-cessing techniques were required to mitigate errors in theraw results and achieve quantitative chemical accuracyfor bonding energies. This work also simulated diazene(H N ) isomerization energies for converting cis-diazeneto trans-diazene, marking the first time a chemical transi-tion has been modeled on a quantum computer. There-fore, despite the limitations from noise, digital simula-tions on SC QPUs show promise for chemical simulations. III. QUANTUM COMPUTING WITHTRAPPED IONS
Trapped ions [37, 38] were one of the first platformsproposed for building a quantum computer as they forma natural representation of an ideal qubit: all ions areidentical by nature, their high degree of isolation fromthe environment leads to excellent coherence times andinteraction with radio-frequency (rf) and laser light al-lows for high-fidelity gate operations. Qubits are encodedin the electronic states of individual ions trapped by elec-tric fields in an rf Paul trap. Two-dimensional traps canbe micro-fabricated on silicon chips, called surface traps,and can contain multiple trapping and interaction zonesas well as integrated microwave and laser access [39–42]. Interaction of the electronic states of neighbouringions is negligibly small [43], but ions are strongly cou-pled via their motion which can be exploited to createentanglement between different ions necessary for multi-qubit gates [44]. Ions are confined in long chains, withinwhich all ions can interact with each other. Chains canbe split and merged, and ions can be moved across thechip between different zones, providing large flexibilityof connections [45–47]. Many different elements are usedas ion species, but all ions are typically singly-chargedand have a single remaining valence electron. Popularchoices of ion are Yb + , Ca + and Be + [48–52]. Qubitstates can either both be encoded in ground-state lev-els (hyperfine- and Zeeman-qubits [53, 54]) with tran-sition frequencies in the rf range, or with the excitedstate encoded in a meta-stable state (eg. D / ) leadingto optical transition frequencies [55]. Different propertiesof the atomic species affect the qubit performance. Forexample, some hyperfine qubits are robust to magneticfield noise, which is the main source of decoherence intrapped ion qubits, and therefore have greatly enhancedcoherence times [53, 56]. Other important factors arethe existence of low-lying D manifolds, which can assistreadout but cause errors due to scattering in laser gateoperations; the ions’ mass where lighter ions allow fastergates; excited state lifetimes for optical qubits; and tran-sition frequencies depending on the availability of suit-able lasers. Scaling up devices from tens to thousands or millionsof qubits is arguably the biggest challenge in realising aquantum computer. The trapped ion community pur-sues several paths towards scalability. In the quantumcharge-coupled-device (QCCD) architecture [37, 57] ionchains are broken up into smaller groups in individualzones, instead of forming a single long string. For scala-bility beyond a single chip proposals include connectingseparate traps via photonic links [58–60] and shuttlingof ions across arrays of chips [61]. Another importantingredient for scalability is the simultaneous use of dif-ferent ion species, which allows sympathetic cooling ofions without affecting the electronic state of the logicand memory qubits [62] and better spectral isolation forion-photon entanglement. Strings of ions can be split,merged and shuttled between different zones with negli-gible effect on the spin state and coherence, but a slightincrease in ion temperature [45–47]. While ion traps canbe operated at room temperature, their performance isenhanced at cryogenic temperatures due to a reductionin heating rate and an increase in ion lifetime. Coolingdown to ∼
10K with liquid helium cryostats suffices forthis purpose.Trapped ions have the longest coherence times of allcontending platforms for building a quantum computer.Even though their individual operations are slower thanin solid state systems, they still possess a superior ratio ofgate operation time to coherence time, which ultimatelyresults in record single- and two-qubit gate fidelity. Whiletechnology and infrastructure for solid-state systems ismore mature than laser technology due to developmentsmade for classical computer chips, rapid progress in thestability, miniaturisation, and integration of laser andion trap systems has been achieved in the last few yearsdue to the influx of resources and increase in demand.Trapped ion quantum computers also benefit from theabsence of noisy direct environments which are presentin solid state systems, and the high degree of connec-tivity and flexibility of connections in trapped ion sys-tems. Remaining challenges are to reduce gate errors forlarger numbers of qubits, which tend to increase with thenumber of ions, and to improve automatisation, robust-ness and crosstalk for building larger devices. Furtherresearch is also required in trap fabrication, as one of themajor gate error sources stems from anomalous heatingof the ion crystals, thought to be caused by surface effectson the ion trap electrodes [63–66].
A. Qubit readout, initialisation and cooling
Qubits are read out via state-dependent fluorescencedetection. All ion species used for quantum computinghave a short-lived excited state that predominantly de-cays back into the qubit ground state manifold. For opti-cal qubits and some hyperfine qubits the qubit frequencyis sufficiently large that the fluorescence laser only cou-ples to one of the qubit states, the ‘bright’ state. To-gether with selection rules preventing decay from theexcited state into the opposite ‘dark’ qubit state, thisallows direct fluorescence readout. For qubits withoutdirect state selectivity of the fluorescence laser, the darkstate is transferred into a ‘shelf’ state that does not cou-ple to the fluorescence laser and the excited state. Ion-position resolved fluorescence can be detected with ar-rays of photomultiplier tubes or avalanche photodiodes,on an electron-multiplying charge coupled device cam-era [67], or with superconducting nanowire single-photondetectors integrated into the trap chip [68]. Fluores-cence can be collected over a fixed time-bin and anal-ysed with threshold or maximum likelihood algorithms,or with real-time analysis and adaptive readout duration.With sufficiently low background counts and high pho-ton collection and detection efficiency, real-time analysisachieves the same fidelities as fixed-time threshold anal-ysis, but is considerably faster [68, 69].Qubit initialisation is performed via optical pumping,using the same excited states as for fluorescence read-out. Either frequency or polarisation selectivity is usedto ensure that population is excited out of all groundstates apart from the target initial state. Different statescan be prepared by applying a sequence of single qubitoperations after optical pumping.For optimum gate fidelities ion crystals need to becooled close to their motional ground states, which isperformed with laser cooling. Typically ions are con-tinuously Doppler cooled during idle time. Before anexperiment resolved-sideband cooling (RSBC) is usedto further cool relevant motional modes to an averagemotional mode occupation of ¯n (cid:46) .
1. Alternativelyelectromagnetically-induced transparency cooling can beused to cool all modes simultaneously [70]. While consid-erably faster than RSBC, especially for larger ion strings,the final temperature reached is slightly higher.
B. Qubit manipulation
Single qubit gates can be driven directly using rfin Zeeman- and hyperfine-qubits, or using a narrow-linewidth laser to drive the quadrupole transition in op-tical qubits. Alternatively a pair of lasers which are fardetuned from the excited state and have the qubit fre-quency as their frequency difference can be used to drivequbit rotations via two-photon Raman transitions. Ro-tations around the z-axis can be performed trivially bypropagating the phase of all future operations. The phaseis defined by a direct digital synthesis frequency sourcethat is either applied directly on the ions as rf or controlsthe frequency, amplitude and phase of the laser beams viaan acousto-optic modulator (AOM). Rf operations coupleonly very weakly to the motion due to their low photonenergy and can already be performed at Doppler-cooledtemperatures at very high fidelities [53]. They also havesuperior phase stability compared to lasers and can eas-ily be integrated into surface traps, but are harder to address onto single ions.Multi-qubit gates create entanglement between differ-ent qubits and require the ions’ motion as a bus of in-teraction between the ions. There are different schemesfor entangling gates, with the most established ones be-ing the closely related Mølmer-Sørensen (MS) gate [71]and the σ z geometric phase (ZGP) gate[72]. Both createa spin-dependent force on the ions; the MS gate in the | + (cid:105) , |−(cid:105) basis and the ZGP gate in the |↑(cid:105) , |↓(cid:105) basis. Thisforce leads to motional excitation and displacement forone spin parity combination (eg |↑↓(cid:105) ) but not the other(eg |↑↑(cid:105) ). Displaced spin states acquire a phase which ul-timately leads to the entanglement. The propagator of atwo-qubit gate with these schemes is diag(1,i,i,1), whichcorresponds to a controlled-PHASE gate. This gate canbe transformed into a CNOT gate via additional single-qubit operations. Both gate mechanisms are first-orderinsensitive to the ion temperature, which makes themmore robust and is an important factor in the high fideli-ties achieved. ZGP gates cannot be performed directlyon the low-decoherence clock qubits, but are insensitiveto the absolute magnetic field offset. Two-qubit gateshave been performed both with lasers [50, 52, 55, 73]and rf [74–77] as well as between ions of different ele-ments [54, 78–80]. Due to the weak motional couplingrf multi-qubit gates are considerably slower than lasergates. Gates can be performed globally on all ions in astring simultaneously or addressed locally to a specificsubset of ions [48, 51]. C. Performance indicators
Coherence times in trapped ions are T ∗ = 50 s onmagnetic-field insensitive clock qubits [53] and reachover an hour by employing dynamical decoupling se-quences [56]. State-preparation and measurement errorsare ε < · − , with a mean duration of 46 µ s [67–69],where ε ≡ − F for fidelity F . Single qubit gateshave been performed with errors ε = 1 . · − foran rf π/ µ s pulse duration on a single ion[53]. Fast single qubit gates can be implemented witha pulsed laser trading off fidelity against speed, achiev-ing ε = 7 · − for t π/ = 40 ps [81]. Two qubit gateerrors are ε = 8(4) · − at a gate time of t g = 30 µ s[49, 50, 52] for laser gates, and ε = 3(1) · − at a gatetime of t g = 3 . ε = 2 . · − in t g = 1 . µ s[82]. D. Use case
Various algorithms have been implemented on trappedion systems, including Shor’s algorithm and Grover’ssearch algorithm [83, 84], demonstrations of error cor-rection [85, 86], analogue quantum simulations, such asthe simulation of many-body dynamical phase transitions
FIG. 2.
IonQ quantum computer based on a chainof trapped ions:
A high-numerical aperture lens allowsboth individual addressing and readout of the ions. A multi-channel AOM is used to modulate the amplitude, frequencyand phase of the individual laser beams. Inset: The qubitsare encoded in the hyperfine ground-states |↑(cid:105) = S / , F = 1and |↓(cid:105) = S / , F = 0 of the trapped Yb + ions. Gateoperations are performed via a two-photon Raman process,coupling to the excited P states (purple and orange beams).Figure adapted from [89]. [87] exceeding the capabilities of classical computers, aswell as several VQE demonstrations [88–90], for exampleestimating the ground state energy of H , LiH and H O.Fig. 2 shows the ion trap quantum computer of IonQ,which is commercially accessible and was used to performVQE on four individually addressable qubits encoded ina string of Yb + ions to estimate the ground state en-ergy of the water molecule [89]. The quantum circuitimplementation for the energy-evaluation was optimisedto take advantage of the asymmetric state measurementfidelities of the |↑(cid:105) and |↓(cid:105) states, and the higher fidelity( ε φ = π/ (cid:46) · − ) of partially entangling gates XX( φ )( φ < π/
2) compared to full entangling gates XX( π/ ε (cid:46) · − ). The optimised circuit comprised 13 CNOToperations and achieved an energy uncertainty close tothe chemical uncertainty of 1 . IV. SILICON QUANTUM COMPUTER
Today’s digital age is enabled by the relentless progressand optimisation of semiconductor materials and tech-nology. From an industrial standpoint, the use of well-established nanofabrication techniques for the develop-ment of quantum machines would be economically attrac-tive to achieve large-scale systems. As discussed, someof these manufacturing techniques are already applied tosuperconducting and ion trap quantum platforms, andare expected to become central for the development ofsilicon-based systems, offering the prospect of integrat- ing millions of qubits on chips at affordable manufactur-ing costs, akin to classical commercial electronics. Be-sides this technological motivation, silicon is a particu-larly suitable material for spin-based quantum devicesfrom a performance viewpoint. Through isotopic purifi-cation, the only isotope bearing a nuclear spin ( Si) innatural silicon can be nearly completely removed, mak-ing the silicon crystal a quasi-spin-noise-free environmentfor the qubit. This results in silicon spin-qubits havingthe longest coherence time among solid-state implemen-tations.Besides silicon, there exists a large variety of semicon-ductor systems currently under investigation for quan-tum computing applications [91–94]. The main differ-ences lie in the type of material (e.g. natural or purifiedsilicon, synthetic diamond, silicon carbide, heterostruc-tures such as GaAs/AlGaAs, Si/SiGe or Ge/SiGe), theoperational conditions (ranging from room temperaturedown to millikelvin temperature), the way each qubit isspatially confined within the material (e.g. gate-definedquantum dots, etched nanowires, atomic-size crystallo-graphic defects, implanted dopant impurities), the waythe qubit state is readout (e.g. electrical readout viacharge sensors, or optical readout through photolumi-nescence), and the way the qubit state is manipulated(e.g. electron spin resonance via magnetic field puls-ing, electric dipole spin resonance via electric field puls-ing). Despite such diversity, a common denominatorin most platforms is the choice of electron/hole spinsas the two-level system embodying the qubits. Theparadigmatic encoding is represented by a single spinin a static magnetic field with its two Zeeman-split en-ergy levels representing the states | (cid:105) and | (cid:105) . Otherimplementations that have been explored include two-electron singlet-triplet qubits, three-electron charge-spinhybrid qubits and three-electron exchange-only qubits.Such rich ecosystem gives rise to significant performancevariations among qubit implementations. The trade-offscan be many, including the robustness to specific noisesources and the ease of operation. The coherence timescan range from few tens of nanoseconds in GaAs/AlGaAsquantum dots to few seconds in silicon dopants, and thesingle-qubit gate time can vary between sub-nanosecondand hundreds of nanoseconds in Si/SiGe quantum dotsand silicon dopants, respectively.In this Section, we are going to focus our attention ona particular type of semiconductor qubit system, whichhas been deployed for the realisation of the first spin-based quantum computer in the Cloud: SPIN-2QPU [95],developed at QuTech (a collaboration between TUDelftand TNO). It consists of two single electron spin qubitsin a double quantum dot (DQD) that is electrostaticallydefined by metallic gate electrodes deposited on top ofan isotopically purified Si/SiGe heterostructure, as illus-trated in Fig. 3 (a) and Fig. 3 (b).Similar to the other quantum processors discussed pre-viously, spin-based machines must meet certain func-tional criteria. These include reliable initialisation to aknown state, high fidelity projective readout of the fi-nal state, and qubit manipulation through high-qualitysingle- and two-qubit gates. Let us see how SPIN-2QPUsatisfies these criteria. A. Qubit initialisation & readout
The readout of the qubit state is ultimately a mea-surement of the electron spin orientation. However, themagnetic moment of a single spin is exceedingly smalland its direct detection quite difficult. By contrast, thedetection of small displacements of single charges is rou-tinely carried out in semiconductor devices. To this end,SPIN-2QPU uses a single-electron transistor (SET) ca-pacitively coupled to the DQD, as shown in Fig. 3 (b).Whenever a single electron leaves/enters the DQD, theSET produces a discrete jump in the value of its electriccurrent caused by a change in its operation point.Reading out the spin state is, therefore, a matter ofmaking a so-called spin-to-charge conversion, wherebythe electron is allowed to tunnel in or out the DQD in away that depends on its spin state, equivalent to whetherthe qubit is in state | (cid:105) or | (cid:105) . As shown in Fig. 3 (c),the selection rule is energy-based. A single spin in one ofthe dots is capacitively coupled to the SET and tunnelcoupled to a reservoir. After spin manipulation, the dot’senergy level is tuned with a gate voltage pulse such thatthe Fermi reservoir lies between the two Zeeman-splitspin states. If the electron is in state |↓(cid:105) , it does not haveenough energy to leave the dot, and there is no SET cur-rent change due to a lack of charge rearrangement. For astate |↑(cid:105) , the electron can tunnel out of the quantum dotand into the reservoir, leading to a change in SET cur-rent until a new electron tunnels in and re-initializes thequbit to its ground state. Current traces for these twoalternative scenarios are shown in Fig. 3 (d). Note thatin this system initialization can be seen as a by-productof readout, given that an electron with a known spin, i.e. |↓(cid:105) , always resides in the dot at the end of the sequence. B. Qubit manipulation
Analogously to other qubit realisations, a spin-qubitrequires independent rotations about the axes of theBloch sphere (single-qubit gate), as well as rotations thatare dependent on the state of another qubit (two-qubitgate), in order to form a set of universal quantum gates.Through a two-qubit gate, entangled states can be cre-ated when one of the two qubits starts in a superpositionof states.SPIN-2QPU carries out single-qubit gate operationsthrough electric dipole spin resonance (EDSR). It con-sists of a microwave modulated electric pulse deliveredthrough a gate electrode that oscillates the electron wave-function. This has the effect of rotating the electron spinwhenever the electron experiences a time-varying mag- N V P u l se ( m V ) -5051010 Load Read Empty N Time (ms)0 2 4 6 8 10 |↓ 〉 |↑ 〉 |↑ 〉 |↓ 〉 LoadEmpty ReadRead (a)(c) (b)(d)
FIG. 3. (a) Schematic cross-sectional view of a DQD deviceused to control two spin qubits. Top metal gates (yellow ar-eas) are used to tune the conduction band profile (dashed line)in the Si layer and isolate two electron spins. An n-type dopedregion of semiconductor (pink shaded area) is used as an elec-tron reservoir tunnel-coupled to the left QD. A layer of cobalt(blue box) is deposited on top of the gate layers to generate acontrolled magnetic field gradient across the DQD. (b) SEMmicrograph of the DQD in (a). The aluminum gates are pat-terned with electron-beam lithography. The two qubits areformed under the gates highlighted with red and blue circles.The SET detector is formed under the gates highlighted inyellow. Gates that accumulate the electron reservoirs for theDQD and the SET are connected to Ohmic contacts and high-lighted by crossed squares. Dashed lines indicate the regionwhere the micromagnet is deposited. (c) Schematic diagramof energy levels for the left QD and the electron reservoir dur-ing the readout pulse sequence. Energy levels in the QD areZeeman-split according to spin polarization. (d) Pulsing se-quence (top) for the spin readout and normalised SET signalfor spin-up (middle) and spin-down (bottom) qubit states.Panels (a) and (b) are adapted from Ref. [95]. Panels (c) and(d) are adapted from Ref. [96]. netic field resonant with its Zeeman splitting. This re-quires the presence of a synthetic spin-orbit field obtainedthrough a local magnetic field gradient in the DQD,which is engineered by depositing a cobalt micromagneton top of the device gate layer (see Fig. 3 (a)). Theamplitude of the EDSR pulse controls the spin vector’srotation frequency around the Bloch sphere, its phasecontrols the rotation axis, and its duration controls therotation angle. The frequency of the pulse allows one toselect which qubit is manipulated, given that each elec-tron experiences a slightly different magnetic field due tothe different position within the DQD.SPIN-2QPU carries out two-qubit gate control viamodulation of the exchange interaction. The idea is toquickly turn on the tunnel coupling between two neigh-bouring spins by applying a gate-voltage pulse that low-ers the tunnel barrier between their corresponding quan-tum dots, so that the electron wavefunctions overlap.Such overlap leads to an exchange interaction betweenthe spins, which can be exploited for conditional gateoperations.
C. Operational conditions & performanceindicators
The readout protocol is effective if the qubit energylevels are separated by at least a few times the thermalenergy. This is ultimately the reason why SPIN-2QPUand similar semiconductor-based quantum systems needto be operated at dilution refrigerator temperature ( T )and in the presence of an external static magnetic field( B ). Typical conditions require B ≈ T ≈
50 mK.The duration of the readout sequence is ultimately de-termined by the tunnelling rate between the DQD andthe reservoir, as well as by the bandwidth of the SET de-tector. SPIN-2QPU’s readout duration is ≈ µ s perqubit and its readout fidelity is approximately 85%.Given a single-qubit gate duration of approximately250 ns and a phase coherence time of at least 6 µ s, SPIN-2QPU achieves single-qubit fidelities in excess of 99.0%.As for 2-qubit operations, the only allowed native gateis CZ. Hence, other gates like CNOT and SWAP have tobe decomposed into CZ operations in combination withsingle-qubit rotations. This comes at the expense of fi-delity and operational time. A detailed benchmark forCZ is ongoing. Preliminary data show gate duration ofaround 150 ns and fidelity in excess of 90%, but this lat-ter figure is likely to be a conservative underestimate atthis stage. D. Use case
At present, we are not aware of VQE simulations car-ried out with SPIN-2QPU or any other semiconductorqubit system, possibly due to the limited qubit count.By contrast, 2D arrays of semiconductor QDs have beenused for analog simulations of magnetic and insulatingmaterials by spatially engineering Hamiltonians onto thearray [97, 98]. It is, however, useful to report that ithas been possible to run digital algorithms of differentkinds (Deutsch–Josza and Grover) on the SiGe processorthat QuTech used to prototype SPIN2-QPU [99]. Thisultimately casts a positive light for future uses of semi-conductor machines in computational chemistry.
V. DISCUSSION
In recent months significant attention has been drawnto superconducting quantum hardware because a team atGoogle achieved a much anticipated milestone, namelyquantum supremacy [100]. By quantum supremacy oradvantage, it is meant that a quantum computer is ableto produce the solution to a computational problem thatwould be otherwise impossible in a reasonable time with a classical machine. Google scientists achieved this witha 53-transmon-qubit processor (Table I) by showing effi-cient sampling of random quantum circuits. While thisresult is of primary importance for the field as a whole,the problem tackled did not bear any relevance to molec-ular chemistry. Therefore, with regard to this type ofproblem, a quantum advantage is yet to be demonstrated.However, in a more recent study [6], the Google teamused the same quantum processor for chemical simula-tions, as discussed in Section II D. They demonstratedthe most complex ground state simulation to date withas many as 24 spin-orbitals mapped onto 12 qubits. Al-though these calculations are relatively straightforwardwith a conventional supercomputer, they represent a sig-nificant advance of the state-of-the-art in quantum com-puting power, as the number of qubits used and orbitalssimulated in prior experiments was no more than six [7].While Google’s quantum hardware is scheduled to be de-ployed onto cloud services imminently, there is alreadya variety of tools made available by Google scientists toexperiment with emulated hardware tailored for applica-tions in molecular chemistry[101]. As for superconduct-ing hardware readily available in the cloud, one has tocurrently turn to IBM or Rigetti, see Table I. IBM hasabout a dozen QPUs in the cloud, arguably the mostextensive offer yet. Just through its Open Access ser-vice, the community can access eight machines with qubitcounts ranging from 1 to 15 and QV ranging from 8 to32. The most powerful QPUs with qubit counts up to65 and QV up to 32 are available for business clientsvia Premium Access. A recent breakthrough has led toQV=64 for a new 27-qubit system not yet available inthe cloud. IBM scientists were among the pioneers in ex-ploiting QPUs for molecular chemistry applications (seeTable I) [7]. More recently, they have also shown thatimproved simulation accuracy can be obtained by adopt-ing error mitigation techniques at algorithmic level [8].This is important because it can be used to enhance thecomputational power of a processor without any hard-ware modification.Quantum machines based on trapped ions have pro-gressed very quickly in the past year alone. While de-vices used for digital quantum computing typically havea lower qubit count than their superconducting counter-parts, analogue quantum simulation has been performedon strings containing up to 53 qubits [87] and singlequbit operations have been performed in devices con-taining up to 79 qubits [102]. Due to superior gate fi-delity and qubit-to-qubit connectivity, the quantum vol-ume of ion trap processors is outperforming supercon-ducting devices even for smaller numbers of qubits. Re-cently, corporate research teams at IonQ and Honeywellhave made QPUs available through the wider cloud ser-vices of Amazon and Microsoft, see Table I. Honeywell’sQPU shows the largest volume to date, i.e. QV=128,whereas IonQ has announced the imminent launch of anupgraded QPU claiming QV=4 . × . Trapped ionmachines have also been used for molecular chemistry0 Manufacturer Platform Cloud access Max
IBM Superconducting IBM QuantumExperience(Open access) 15 (Melbourne) 99.97%, 99.16%(Santiago) 32 (Santiago) H , LiH, BeH ,NaH, KH, RbHIonQ Trapped Ions Microsoft Azureor AmazonBracket 11 99.50%, 97.50% not published H OQuTech Silicon QuantumInspire 2 (Spin2-QPU) ≈ ≈
90% not published noneGoogle Superconducting Google Quan-tum AI 53 (Sycamore) 99.85%, 99.35% not published H N , H , H ,H , H Rigetti Superconducting Rigetti Quan-tum Cloud 31 (Aspen-8) 99.8%, 95.9% 8 (Aspen-4) NaH, H Honeywell Trapped Ions Microsoft Azureor AmazonBracket 10 (H1) 99.97%, 99.5% 128 noneTABLE I. Quantum computing hardware in the cloud. Wherever more than one QPU is available, the relevant machine isindicated within brackets. Simulated molecules column denotes experiments run with any quantum machine from the relevantmanufacturer, not necessarily one of those listed. IBM hardware considered is limited to Open Access services. Google cloudservices are limited to emulators at present, although the reported chemical simulations have been performed with proprietaryphysical hardware. simulations[88, 89, 103]. The most complex molecularsimulation performed to date with trapped ions is theevaluation of the binding energy of the water moleculewith a 4-qubit QPU from IonQ [89], as discussed in Sec-tion III.The 2-qubit silicon quantum processor made byQuTech is the only spin-based system in the cloud. Theservice through which it is accessible, the platform Quan-tum Inspire, also provides a more powerful alternativebased on a 5-qubit superconducting QPU. Silicon SPIN2-QPU has been the latest to be deployed (April 2020) andis not yet fully characterised, hence only approximate fi-delities are quoted in Table I. Although no chemical sim-ulations have been attempted yet, one should expect thatthe semiconductor community will soon fill this gap. Themodest qubit count should not be an insurmountable im-pediment if one considers that early 2-qubit QPUs weresuccessfully used to simulate diatomic molecules [3–5].Undoubtedly, Si-based machines have yet to cover muchground before becoming realistic competitors of the othertwo major platforms. For example, high-fidelity single-and two-qubit gates have only recently been achievedand are not yet on par with those of the other hard-ware platforms [92]. Furthermore, qubit variability dueto atomic level defects in the material and its interfacesis an issue that currently hampers scalability. Nonethe-less, the interest around these devices is justified by thefact that in principle they can be manufactured with in-dustrial CMOS technology, and have the smallest qubitfootprint [104]. This bodes well for future upgrades ofsuch systems towards the million-qubit-machines neededfor useful applications. Finally, note that there existsanother type of silicon QPU based on photonic technol- ogy (as opposed to spins) with two systems accessible viacloud services [105, 106].
VI. CONCLUSION AND OUTLOOK
A lot of theoretical and experimental ground has beencovered since the early 80s, when Feynman proposedto use controllable quantum devices for computationalproblems in chemistry and physics. There are nowdozens of small-scale quantum computers in the cloudand many more in academic and corporate laboratoriesworldwide. The electronic structures of simple moleculesranging from diatomic systems to chains of a dozenatoms has been determined with several QPU incarna-tions.In this Article, we have discussed the hardware ofthe most popular types of quantum computers, forwhich we have summarised the main techniques forphysical encoding, manipulation and readout of quan-tum information. We have paid particular attentionto the machines that the reader could easily access viacloud services, i.e. superconducting-, trapped ion- andsilicon-based processors. For these, we have describedthe main performance specifications and operationalconditions. Our target has been to highlight to whatextent these early prototypes have been employed forchemistry simulations. The underlying message is that,despite relentless progress, none of the machines builtthus far is yet advantageous to a chemist, if compared toclassical computational methods. What needs to happento change this?In order to achieve a sizable quantum advantage1in computational chemistry with NISQ machines, thecoordinated efforts between quantum hardware andquantum algorithm developers will need to continueif not intensify. Hardware improvements in terms ofqubit count, qubit connectivity, quantum gate speedand fidelity, as well as overall QPU volume will be acentral focus for years to come. However, recent break-throughs [6, 8] have shown that tailoring algorithms tothe specific quantum hardware available in combinationwith error mitigation techniques could be important foraccurate chemical computation on near-term machines.Particularly, restrictions to realisable gates inherentto NISQ processors could be bypassed with ad-hoccompilation methods.Beyond the NISQ era, i.e. without today’s limi-tations due to noise, there will be the possibility oftaking full advantage of the computational speed-upof quantum systems. QEC protocols will have to bereliably implemented to produce such step change.During this transition, a risk to be avoided will be thattoday’s capability restrictions, rather than being liftedaltogether, will be merely transferred from the quantumlayer onto the classical control layer [107]. There are twocomplementary considerations to this potential problem.Firstly, QEC will require fast feedback between mea-surement and control, and communication latency maybecome an issue. If there is a sizable physical distancebetween the quantum hardware and the classical controlhardware, which is likely for cryogenic QPUs, delays inthe communication lines may pose a synchronizationchallenge if they become of the same order as the gatetime. Secondly, a computational bottleneck may occur in handling error correction cycles for large number ofphysical qubits. For example, a QPU with a millionqubits corrected with cycles of 1 µ s will require classicalinformation processing at a bandwidth of 1 Tbit/s. Ifboth latency and bandwidth issues are to be solved atonce, novel ultra-low-power cryogenic RAM and CPUmay need to be developed, so that they could sit nearor within the same chip of a cryogenic QPU withoutgenerating detrimental heat loads [104].We believe that the challenges described do notrepresent a fundamental roadblock towards large-scalefault-tolerant quantum computing. However, they dopose significant engineering hurdles that will requiresynergies between quantum and electronic engineers,as well as quantum software developers and end users.We hope that this Article will trigger the curiosity oftheoretical and quantum chemists in trying out theavailable cloud machines, get involved into the ongoingconversation and, eventually, steer quantum systemsdevelopment to the benefit of their scientific agenda. ACKNOWLEDGMENTS
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