Efficient Assessment of Process Fidelity
Sean Greenaway, Frédéric Sauvage, Kiran E. Khosla, Florian Mintert
EEfficient Assessment of Process Fidelity
Sean Greenaway, Fr´ed´eric Sauvage, Kiran E. Khosla, and Florian Mintert
Physics Department, Blackett Laboratory, Imperial College London,Prince Consort Road, SW7 2BW, United Kingdom (Dated: February 17, 2021)The accurate implementation of quantum gates is essential for the realisation of quantum algo-rithms and digital quantum simulations. This accuracy may be increased on noisy hardware throughthe variational optimisation of gates, however the experimental realisation of such a protocol is im-peded by the large effort required to estimate the fidelity of an implemented gate. With a hierarchyof approximations we find a faithful approximation to the quantum process fidelity that can be es-timated experimentally with reduced effort. Its practical use is demonstrated with the optimisationof a three-qubit quantum gate on a commercially available quantum processor.
I. INTRODUCTION
Experimental progress in developing quantum com-puters has led to the realisation of noisy intermediatescale quantum (NISQ) devices in a wide array of exper-imental platforms [1–4]. Whilst the number of qubitsin these devices is approaching that needed for quantumsupremacy [5], their noisiness remains a fundamental lim-iting factor in the development of useful applications [6].As such, a number of techniques have been developed for error mitigation in quantum computations, wherein ad-ditional measurement data and classical post-processingare used in order to extract relatively noise-free resultsfrom the noisy devices [7–10].Many of these techniques have focused on obtaining ac-curate expectation values from noisy devices [11–16] andare most commonly paired with variational quantum al-gorithms (VQAs) [17–22], hybrid quantum-classical algo-rithms in which a parameterised ansatz is updated usingexperimental measurements in order to optimise for somerelevant observable. An additional, complementary errormitigation strategy which could be immensely powerfulwould be to variationally optimise quantum channels byadapting them from their textbook implementation suchthat the optimised channel more closely realises the de-sired operation [23].In order to perform channel optimisation, a figure ofmerit which characterises the quality of an implementedchannel efficiently is required. The natural choice for thisis the process fidelity [24, 25], however it is very expensiveto evaluate experimentally. Instead of evaluating the pro-cess fidelity directly, it is far more efficient to estimate itin terms of a relatively small number of expectation val-ues independent of the system size [26, 27]. The fact thatthe estimation is unbiased with a relatively low varianceis sufficient for channel optimisation: techniques fromquantum optimal control which have been shown to behighly effective in similar settings [28–32] may be appliedhere.Unfortunately, whilst estimates of the process fidelityare theoretically efficient, their implementation on realhardware necessitates changing the experimental setupat every shot of the experiment. With the limited speed at which NISQ hardware may be controlled and accessed,this translates to a substantial decrease in efficiency, pre-venting the estimation protocol from being useable inoptimisation.An alternative figure of merit which may be imple-mented using only a small number of unique experimen-tal settings is thus desirable. In this work we introduce ahierarchy of approximations to the process fidelity whichwe refer to as k − fidelities . These are given in terms of aphysically implementable set of expectation values which,together with the fact they are approximately monotonicfunctions of the process fidelity, means they have the po-tential to provide alternative figures of merit by whichthe quality of quantum channels may be assessed.In particular, the leading order term, the 0 − fidelity,is especially useful since it satisfies the requirement ofbeing efficiently estimable with few unique experimentalsettings. The 0 − fidelity is a faithful approximation tothe process fidelity, that is, it is maximised if and onlyif the process fidelity is also maximised. This makes it asuitable figure of merit for the optimisation of quantumchannels. We find that the 0 − fidelity not only approx-imates the process fidelity well, particularly at high fi-delities, but we also find that the approximation becomesbetter as the system size increases, which we demonstratenumerically.The key advantage of the 0 − fidelity over the processfidelity is that it can be efficiently estimated even underthe constraints imposed by NISQ platforms, allowing theestimation protocol to be repeated multiple times as ne-cessitated by an optimisation routine. We demonstratethe superior performance of the 0 − fidelity estimationsunder these conditions both numerically and through ex-periments performed on an IBM quantum device. II. THE 0 − FIDELITYA. Evaluating the Quality of Quantum Channels
Any attempt to realise a desired channel Λ that mapsinput states ρ I to their designated output states ρ f =Λ( ρ I ) experimentally will inevitably result in the realisa- a r X i v : . [ qu a n t - ph ] F e b tion of a channel Γ that does not perfectly coincide withΛ. The similarity between Γ and Λ is typically quantifiedby the process fidelity , F (Λ , Γ) = 1 d d (cid:88) i =1 Tr[Λ( σ † i )Γ( σ i )] , (1)expressed in terms of a complete set of mutually or-thonormal operators σ i on a d -dimensional Hilbert space.In order to assess the process fidelity experimentally, itis essential that these inputs be quantum states, i.e. Her-mitian, positive semi-definite operators. However, whilea complete set of operators contains d elements, thereare only d mutually orthogonal quantum states.A formulation of the process fidelity that is consistentwith the requirement that the channel inputs be quantumstates is given by F (Λ , Γ) = 1 d d (cid:88) ij =1 [ B − ] ij Tr[Λ( ρ i )Γ( ρ j )] , (2)with the matrix B ij = Tr[ ρ † i ρ j ] comprised of the mutualoverlaps of the states ρ i . The change in notation from σ i to ρ i emphasises the experimentally motivated restrictionto quantum states.The term Tr[Λ( ρ i )Γ( ρ j )] in Eq. (2) can be understoodas the expectation value of the observable Λ( ρ i ) with re-spect to the state Γ( ρ j ), i.e. the state obtained with theevolution described by the channel Γ after initializationin the state ρ j . For most channels Λ and most sets ofstates ρ i , however, the observables Λ( ρ i ) have entangledeigenstates, and thus this expectation value is imprac-tical to measure experimentally. It is thus necessary toexpress the process fidelity in terms of a complete set ofmutually orthonormal, local observables W j as F (Λ , Γ) = 1 d d (cid:88) ij =1 C ij Tr[Γ( ρ i ) W j ] , (3)with C ij = d (cid:88) l =1 [ B − ] li Tr[Λ( ρ l ) W j ] . (4)The full experimental protocol entailed by Eq. (3) impliesthe preparation of d initial states ρ i and the measure-ment of d observables W j per initial state for a total of d experimental settings. Since the dimension d grows expo-nentially in the number of qubits, the experimental effortrequired to evaluate the process fidelity is prohibitivelyhigh even for a moderate number of qubits.It is, however, possible to estimate the process fidelityusing far fewer experimental settings [26, 27]. The proce-dure involves sampling a small subset of input states andmeasurement bases from a joint probability distributionwhich guarantees the resulting estimates will have a low variance regardless of the specific channels being evalu-ated.Unfortunately, this protocol may only be applied tothe process fidelity as expressed in Eq. (1). The formof Eq. (3) precludes the definition of an estimator withsimilarly favourable statistical properties. The estima-tion protocol thus implicitly relies on the inputs being or-thonormal, meaning that not all of them can be quantumstates. An experimental protocol may still be obtainedby sampling quantum input states from the eigenbasis ofeach σ i input on a shot-by-shot basis, however this relieson the ability to vary the experimental setup at everyshot of the experiment.Given the operation speed of current laboratory con-trol software this can reduce the repetition rate of an ex-periment substantially. Consistently with this, the inter-faces to currently available NISQ devices limit the num-ber of initial states and measurement settings that can beexplored, whereas they do not impose comparably severelimitations to the repetition of the same experiment i.e. with the same initial state and measurement basis [33].In NMR quantum computing [34–37], the situation iseven more drastic, since expectation values are obtainedfrom the simultaneous measurement of an ensemble ofqubits rather than through individual projective mea-surements [38]. Varying the initial states and observ-ables thus necessarily increases the overhead by a factorof the number of chosen experimental settings, severelyhindering the efficiency of the protocol.The goal of this work is to develop an alternative for-mulation for assessing the quality of implemented quan-tum channels based on Eq. (3) that may be efficientlyestimated without resorting to frequent changes in ini-tial state preparation and final measurement basis. B. The hierarchy of k − fidelities In order to use the process fidelity as in Eq. (3), a setof states which span the space of linear operators mustbe specified. The natural choice is to take these statesto be as close to orthogonal as possible, which may beachieved by minimising (cid:80) i (cid:54) = j Tr[ ρ i ρ j ]. For a single qubit,the analytical solution to this is any set of four stateswhich form the vertices of a regular tetrahedron centredat the origin of the Bloch sphere.In principle equivalent sets of states may be obtainedfor multiple qubits, however in order to minimise un-necessary entangling gates it is important to guaranteethe states ρ i are product states. This may be achievedby constructing multi-qubit states as tensor products ofthese single qubit states. Whilst this means that thestates no longer satisfy the minimisation above, this is asmall price to pay for keeping input states separable.With this set of states defined, the inverse of the over-lap matrix B ij is readily obtained. The inverse of theoverlap matrix B (1) of a single qubit reads (cid:2) B (1) (cid:3) − = − A , in terms of the matrix A with elements A ij = (1 − δ ij ). In the case of n qubits, the inverse of B isthe n -fold tensor product ( − A ) ⊗ n . Collecting termswith a given number of factors of A , this is expressed as B − = n (cid:88) k =0 ( − k Ω k , (5)withΩ k = (cid:88) π ⊗ ⊗ . . . ⊗ (cid:124) (cid:123)(cid:122) (cid:125) k ⊗ A ⊗ A ⊗ . . . ⊗ A (cid:124) (cid:123)(cid:122) (cid:125) n − k , (6)where the sum is performed over all inequivalent permu-tations of identity and operators A .By truncating Eq. (5) at different values of k , one candefine a hierarchical series of k − fidelities . Truncatingthe expansion at the k − th term means that only pairs ofinput states differing by at most k single qubit states willhave non-zero contributions to the k − fidelity; thus the n − qubit process fidelity is equivalent to the n − fidelity.The lower the overlap between ρ i and ρ j , the less thatpair of states contributes to the overall k − fidelity. Thecoefficient c m = n (cid:88) j = m ( − j + m (cid:18) (cid:19) j (cid:18) n − mj − m (cid:19) = ( − m n − m n quantifying this contribution for a pair of states differingby m single qubit states decreases exponentially in m .Thus the higher the order k in Eq. (5), the lower thecorresponding coefficient in the k − fidelity. The leadingorder term, the 0 − fidelity ,˜ F (Λ , Γ) = 1 d d (cid:88) ij =1 Tr[Λ( ρ i ) W j ] Tr[Γ( ρ i ) W j ] , (7)may then be used as an approximation to the processfidelity with highly convenient properties which will beshown in the following section. Moreover, the 0 − fidelitymay be efficiently estimated using a small number ofunique experiments, without requiring expensive shot-by-shot changes to the input state and measurement ba-sis. C. Properties of the 0 − Fidelity
There are a number of properties that the 0 − fidelitysatisfies which make it an effective proxy for the processfidelity:(i) Faithfulness : It is faithful, that is the 0 − fidelity ismaximised if and only if the process fidelity is alsomaximised, i.e. if the two channels being consid-ered are identical.(ii) Monotonicity : It is an approximately monotonicfunction of the process fidelity, meaning that highfidelity channels give rise to high 0 − fidelities, withonly small deviations from monotonicity which de-crease at high fidelities. . . . . Process Fidelity (F) . . . . . . k − F i d e li t y ( ˜ F k ) k = 0 k = 1 k = 2 k = 3 10 − − − | − F | − − − | F − ˜ F k | FIG. 1. Plot of k − fidelity against process fidelity for 10000numerical evaluations of randomly generated 3 qubit unitarychannels compared with a random unitary target. As theorder k increases, the k − fidelity more closely approximatesthe process fidelity, with the 3 − fidelity being equal to it. Athigh fidelities, all orders give rise to approximately monotonic,close approximations to the process fidelity, with the devia-tions becoming very small at very high fidelities as shown inthe inset. (iii) Scalability : Finally, as the system size increases,the 0 − fidelity becomes an increasingly better ap-proximation to the process fidelity.In the following section these properties will be shownthrough analytical proof for (i) and through numericalevidence for (ii) and (iii).The 0 − fidelity may be understood as the average statefidelity between output states Γ( ρ i ) and Λ( ρ i ); each statefidelity is maximised if and only if the states are identical,thus the 0 − fidelity is maximised if and only ifΓ( ρ i ) = Λ( ρ i ) ∀ ρ i . (8)The input states form an operator basis, thus the linear-ity of quantum channels implies the relationΓ( O ) = Λ( O ) (9)for any arbitrary operator O , therefore the 0 − fidelity ismaximised if and only if the two channels are identical.Fig. 1 shows how well each order of the k − fidelity ap-proximates the process fidelity in the assessment of threequbit quantum channels, generated numerically by com-paring 10000 randomly generated unitary channels witha fixed random unitary target channel. All the ordersconverge to the process fidelity at high values, as canbe seen in the inset. The process by which the randomunitaries are generated is outlined in the Appendix. . . . . Process Fidelity (F) . . . . . . − F i d e li t y ( ˜ F ) − − − | − F | − − − | F − ˜ F | FIG. 2. Plot of 0 − fidelity against process fidelity for 10000numerical evaluations of randomly generated 2, 3 and 5 qubitunitary channels evaluated against random unitary targets.The 0 − fidelity approximates the process fidelity increasinglywell as the number of qubits increases. The inset shows that athigh fidelities the 0 − fidelity converges to the process fidelityfor all system sizes. For orders k < n , the k − fidelities are not true mono-tonic functions of the process fidelity, however they are approximately monotonic (this may be seen in the thick-ness of the evaluated points). The 0 − fidelity, as the low-est order, deviates most substantially from monotonic-ity. Nevertheless it is close to monotonic for fidelitiesabove ∼
50% and even at low fidelities the deviationsfrom monotonicity are relatively small, meaning that the0 − fidelity should still be useful as an approximation tothe process fidelity for quantum channel optimisation.As the number of qubits increases the process fidelity isincreasingly well approximated by the 0 − fidelity (Fig. 2).The n = 5 qubit case (green) consistently gives rise to0 − fidelity values which are close to the process fideli-ties, and additionally exhibits the desired monotonicityabove ∼
15% whilst the n = 2 qubit case (blue) onlyexhibits this above ∼ − fidelity may be relied upon as a process fidelityapproximation even for relatively large systems. III. ESTIMATION OF THE − AND PROCESSFIDELITIESA. Estimating the 0 − Fidelity
As with the process fidelity, evaluating the 0 − fidelityinvolves a number of experimental settings scaling expo- nentially with the system size and thus for a practicalexperimental prescription it is necessary to estimate itusing random sampling [26, 27]. The protocol works bychoosing input states and measurement bases from thejoint probability distributionPr( i, j ) = 1 d Tr[Λ( ρ i ) W j ] . (10)These selected settings may then be used to construct anestimator X ( i, j ) = Tr[Γ( ρ i ) W j ]Tr[Λ( ρ i ) W j ] , (11)in terms of experimentally measurable quantitiesTr[Γ( ρ i ) W j ]; the expectation value of X over i and j is equal to the 0 − fidelity.The crucial factor which makes this procedure ex-tremely useful lies in the sample variance of the estima-tor, Var( X ) = 1 − ˜ F , (12)which is bounded by 1 independently of the specific chan-nels being assessed. This means that by taking the meanof l evaluations of X ( i, j ) sampled from Eq. (10) an un-biased estimate of the 0 − fidelity may be obtained witha sample variance of, at worst, 1 /l . Manageable sam-ple variances may be achieved by evaluating a relativelysmall ( l ∼ X ( i, j ) given in Eq. (11), areobtained exactly, however the accuracy of these terms islimited by experimental constraints. In platforms suchas superconducting qubits, the accuracy of estimating X ( i, j ) is limited by the number of experimental repe-titions (shots) m required to acquire expectation valuesfrom projective measurements, a number which dependson the details of the channel Λ and which scales in theworst case as O ( d ). In NMR platforms, where expec-tation values are obtained from a single experiment, theprimary limiting factor is the finite measurement acquisi-tion time which limits the resolution at which expectationvalues may be extracted.The two distinct platforms entail slightly different im-plementations. In both cases, l experimental settings( i, j ) corresponding to expectation values Tr[Γ( ρ i ) W j ] areselected according to the probability distribution Eq (10).In projective estimation , the expectation values are thenobtained experimentally by running each setting m timesand taking the average of the projective measurements,whilst in full trace estimation , the expectation valuesare obtained exactly from a single experimental measure-ment.For estimations of the 0 − fidelity the distinction be-tween the two implementations is merely a technical de-tail, however for estimating the process fidelity the choiceof experimental platform can have a severe impact on thequality of the estimations. B. Estimating the Process Fidelity
In order to compare the quality of the estimation pro-tocol for the 0 − fidelity to the equivalent protocol for es-timating the process fidelity, it is necessary to expandEq. (1) in a local orthonormal basis to obtain an expres-sion which is analogous to Eq. (7) as F (Λ , Γ) = 1 d d (cid:88) ij =1 Tr[Λ( σ † i ) W j ] Tr[Γ( σ i ) W j ] . (13)Since the input operators in Eq. (13) are not quan-tum states, an additional step is required in order toobtain experimentally realisable settings. Once a mea-surement setting corresponding to an expectation valueTr[Γ( σ i ) W j ] has been chosen, an experimental prescrip-tion may be obtained by expanding σ i in its eigenbasis.This results in the relationTr[Γ( σ i ) W j ] = d (cid:88) k =1 λ σ i k Tr[Γ( | φ σ i j (cid:105)(cid:104) φ σ i k | ) W j ] , (14)where | φ σ i k (cid:105) are eigenstates of σ i with correspondingeigenvalues λ σ i k . An appropriate choice of operators σ i (for example, the set of tensor products of normalisedPauli operators) means these states will be separable, andthus can be implemented experimentally. This implies anincrease in the number of expectation values which needto be experimentally evaluated by a factor of d .In projective estimations it is possible to obtain esti-mates of the process fidelity using the same total numberof experiments lm as that required for 0 − fidelity esti-mation by sampling experimental input states from theeigenbasis of σ i on a shot-by-shot basis [26, 27]; as dis-cussed in Sec. II A however, this results in inefficiencieswhich render the strategy inapplicable on NISQ devices.Nevertheless, projective estimations of the process fi-delity may still be obtained using the same number ofexpectation values l and the same total number of ex-periments lm as the 0 − fidelity. For any expectationvalue Tr[Γ( σ i ) W j ], implementing all d eigenstate ex-pectation values Tr[Γ( | φ σ i j (cid:105)(cid:104) φ σ i k | ) W j ] with each experi-ment repeated m/d times is equivalent to estimatingTr[Γ( σ i ) W j ] using m shots. Thus the total number ofexperiments remains the same between the process and0 − fidelity estimations, with the caveat that estimatingthe process fidelity requires dl unique experiments ascompared with only l for the 0 − fidelity.In the case of full trace estimation, each unique ex-periment is only repeated once, thus the total number ofexperiments is l . In this case, implementing all d eigen-states of each σ i necessarily entails a factor of d increasein experimental overhead over estimating the 0 − fidelity. − . − . . . . Estimation error F r e q u e n c y Process Fidelity0 − Fidelity
FIG. 3. Statistical deviations from their true values for 10000numerically evaluated full trace estimations of the process and0 − fidelities, each using l = 160 expectation values per evalua-tion. The estimations were made using the overlap between arandom three qubit target unitary and a perturbed test uni-tary, with the same setup for both the process and 0 − fidelityestimations. The variance of the 0 − fidelity estimations is sig-nificantly lower than those of the process fidelity, making itfar more suitable for an optimisation cost function. C. Comparing the Process and 0 − FidelityEstimates
In this section the quality of process and 0 − fidelityestimations are compared in both the full trace and pro-jective experimental situations. In line with current im-plementations of NISQ devices, in the following numeri-cal simulations the maximum number of unique experi-ments is limited to a maximum value of 900. Details onthe generation of random channels can be found in theAppendix.Fig. 3 shows deviations of the full trace estimates ofthe process (blue histogram) and 0 − fidelities (green his-togram) from their true values, based on a randomlygenerated three qubit unitary target channel and a per-turbed test channel. The estimate of the 0 − fidelity isrealized in terms of l = 160 settings Tr[Γ( ρ i ) W j ], witheach expectation value evaluated exactly. The process fi-delity is sampled with l = 20 settings Tr[Γ( σ i ) W j ] for all d = 8 eigenstates Tr[Γ( | φ σ i j (cid:105)(cid:104) φ σ i k | ) W j ] evaluated exactlyper setting, for a total of dl = 160 experimental settings.Fig. 3 exemplifies the fact that the statistical fluctua-tions in estimates of the 0 − fidelity are much smaller thanthose of the process fidelity, with the standard deviationof the 0 − fidelity estimation errors being ∼ .
05 comparedto ∼ .
16 for the process fidelity.For the case of projective estimations, the number ofshots m may be adjusted such that the total number ofexperiments required for estimating the process fidelityis the same as that required to estimate the 0 − fidelity.Even in this favourable setting for the process fidelity,it is still substantially outperformed by the 0 − fidelityestimates.Fig. 4 shows the standard deviations of the estimation Total experiments . . . . . S t a nd a r dd e v i a t i o n Process Fidelity0 − Fidelity
FIG. 4. Plot of the standard deviation of process and0 − fidelity projective estimations as the number of experi-ments increases. The data are generated numerically through10000 estimations per data point of the comparison between apair three random unitary channels, with the same pair usedto generate all data points and where the number of expecta-tion values l was approximately equal to the number of shots m . The 0 − fidelity standard deviations are consistently belowthose of the process fidelity estimates regardless of the totalnumber of experiments used. Moreover, the process fidelitystandard deviations seem to saturate to a minimum value,whereas no such saturation can be observed for the 0 − fidelity. error for 10000 estimations of the process and 0 − fidelitiesof a randomly generated pair of three qubit unitary chan-nels (with the same pair being used for all data points)as the total number of experiments lm increases. Thedetails for the specific generation of the random chan-nels, along with the experimental parameters l and m used are outlined in the Appendix. The 0 − fidelity esti-mates have much lower standard deviations than thoseof the process fidelity at every allocation of (numericallygenerated) experiments investigated.A choice of l ≈ m expectation values and measurementsettings gives rise to estimates with the lowest variancefor a given total number of experiments lm . Since forthe process fidelity the measurement of l expectation val-ues implies the implementation of dl unique experiments,the maximum number of expectation values which can bemeasured is limited to l ≤ (cid:98) /d (cid:99) , meaning that if onewants to use a relatively high number of experiments inorder to obtain an estimate, it is necessary to use a subop-timal allocation of experiments l < m . This is reflected inFig. 4, which shows that the standard deviations for theprocess fidelity seem to saturate to a minimum value oncethe maximum number of unique experiments is reached.Whilst an equivalent limit given by the finite number ofavailable unique circuits will eventually be reached forthe 0 − fidelity, this limit is higher than that constrain-ing the process fidelity by a factor of d and, moreover, isindependent on the size of the channel being evaluated.The superior performance of the 0 − fidelity estimates isalso reflected in real experimental data. Fig. 5 shows 50estimations of the process (blue crosses) and 0 − fidelities Evaluation . . . . . . E s t i m a t e d v a l u e Process Fidelity0 − Fidelity
FIG. 5. Plot showing 50 estimations of the process and0 − fidelities for a single random three qubit target unitaryimplemented on the IBM Q Toronto quantum device. Themean for the 0 − fidelity (orange line) is slightly higher thanthat of the process fidelity (blue line), since the 0 − fidelityoverestimates the process fidelity as shown in Fig. 2, howeverthe standard deviation of the 0 − fidelity (orange shaded re-gion) is substantially lower than that of the process fidelity(blue shaded region). (orange triangles) for a random three qubit quantum cir-cuit implemented on the IBM Q Toronto quantum com-puter (where the circuit would perfectly implement thetarget channel in the absence of noise). The random cir-cuit used is given in the Appendix.The 0 − fidelity estimations were performed using l =336 circuits (that is, 336 unique experimental settings)each using m = 336 shots, which may be evaluated effi-ciently enough to permit an optimisation involving ∼ dl = 896 circuits,corresponding to l = 112 expectation value evaluationsTr[Γ( σ i ) W k ]) and setting the number of shots to m = 144such that the total number of experiments was equal tothat used for the 0 − fidelity estimations.The 0 − fidelity data are clustered much closer togetherthan the process fidelity, indicating that estimates of the0 − fidelity are more suitable as an efficient protocol forevaluating the quality of quantum channels. Althoughit is not feasible to experimentally perform enough esti-mates to fully capture the statistics for the distributionsas in the numerical analysis above, the data clearly in-dicate that the 0 − fidelity estimations have a much lowerstandard deviation than the process fidelity estimates,with the standard deviation of the 0 − fidelity estimationsbeing ∼ .
033 as compared with ∼ .
07 for the processfidelity. U U U U U U FIG. 6. Circuit diagram for the implementation of a parame-terised CNOT gate between the first and third qubits, whichare assumed to not be physically connected in the device.Each of the U gates has three parameters over which theoptimisation may be performed - the ideal gate may be im-plemented on a noise-free device by setting all the parametersto 0. IV. GATE OPTIMISATION
Estimates of the 0 − fidelity are a highly efficient way ofevaluating the quality of noisy quantum channels. Oneapplication for this is in the variational optimisation ofsuch channels, in which the parameters of a parame-terised channel are varied according to some classical op-timisation algorithm until the 0 − fidelity is maximised.In the following section the results of optimisationsperformed using Bayesian optimisation (BO) [39] are pre-sented. BO is highly efficient and resilient to noise,as demonstrated in its successful application in relatedproblems in quantum optimal control [28–30, 40–43]. Forthe interested reader, a thorough review of BO may befound in Refs [44–46].The target channel in these optimisations was a CNOTgate between non-connected qubits, a three qubit chan-nel necessitated by the limited connectivity of NISQ de-vices [47]. The ideal channel may be implemented usingonly CNOT gates, however a parameterised version maybe generated by appending and prepending single qubitgates on all qubits as seen in Fig. 6. A. Optimisation Results
The results of the optimised circuits based on Fig. 6 areshown in Fig. 7 in terms of the process fidelity evaluatedusing all d measurement settings in Eq. (3). The ac-tual optimisations were performed using estimates of the0 − fidelity however evaluating the final channels throughthe fully evaluated process fidelity allows for direct com-parisons with previous work. This also provides furtherevidence that the 0 − fidelity is an excellent proxy for theprocess fidelity in this setting.The optimised circuits (orange triangles) consistentlyachieved substantially higher fidelities than their unopti-mised counterparts (blue circles) over every experimen-tal run. The average process fidelity for the unoptimisedruns was 0 .
65, whilst the optimisation yielded circuitswith average fidelities of 0 .
76, a significant ( ∼ Optimisation Run . . . . . . . P r o ce ss F i d e li t y UnoptimisedOptimised
FIG. 7. Plot showing the improvement obtained by theBayesian optimisation of the circuit in Fig. 6 for 10 optimi-sation runs given in terms of the process fidelity evaluatedusing all d measurement settings, implemented on the IBMQ Singapore quantum device. Each optimisation run was per-formed using 140 iterations using 0 − fidelity estimations ob-tained using 160 unique circuits, each repeated for 2048 shots,with the final results evaluated using 4096 circuits, each using8192 shots. The optimised results consistently outperform theunoptimised results, demonstrating the effectiveness of usingthe 0 − fidelity as the figure of merit for quantum channel op-timisation. ative improvement.The ultimate goal of optimising quantum channels isto use them as part of a larger algorithm. As such, itis critical that this increase in fidelity is retained whensuch a composition is performed. The results shown inFig. 8 confirm that this is the case: here the experimentalchannel was the CNOT applied three times, which shouldbe equivalent to a single CNOT in the absence of noise.Once again, the optimised circuits attain higher fidelitiesin every experimental run, substantially outperformingthe textbook implementations with average fidelities in-creasing from 0 .
11 to 0 . V. CONCLUSIONS
NISQ devices, whilst having much potential are lim-ited by their inherent noisiness. Quantum channels maybe directly optimised such that the resulting channel pro-duces a much more faithful implementation of the desireddynamics. In order to perform such an optimisation,a figure of merit is required which can efficiently char-acterise the quality of an implemented channel. Direct
Optimisation Run . . . . . P r o ce ss F i d e li t y UnoptimisedOptimised
FIG. 8. Plot showing the process fidelity (evaluated usingall d measurement settings) when the optimised and unop-timised channels are applied three times (equivalent in theabsence of noise to a single application) for 6 runs of theBayesian optimisation of the circuit in Fig. 6 implementedon the IBM Q Singapore quantum device. Each optimisationrun was performed using 140 iterations using 0 − fidelity esti-mations obtained using 160 unique circuits, each repeated for2048 shots, with the final results evaluated using 4096 circuits,each using 8192 shots. The optimised results are substantiallybetter across all experimental runs, with the optimisation in-creasing the average process fidelity from 0 .
11 to 0 . process fidelity estimation would be a natural choice forthis, however its implementation on NISQ hardware isrendered impractical through the requirement that theinput state and measurement basis be changed at everyshot of an experiment.In this work we present an alternative figure of meritwhich overcomes this issue. The leading order term ina hierarchical series of k − fidelities, the 0 − fidelity, is afaithful approximation to the process fidelity which canbe estimated on any quantum platform using only a smallnumber of unique experiments that does not scale withsystem size. Estimates of the 0 − fidelity substantiallyoutperform estimates of the process fidelity under theconstraints imposed by current implementations of NISQdevices, with this advantage being demonstrated bothnumerically and experimentally.The 0 − fidelity is an excellent figure of merit for thedirect optimisation of quantum channels. This is demon-strated through the successful optimisation of a CNOTchannel on an IBM Q device, for which we report sig-nificant ( ∼ − fidelity-based optimisation routinecould be applied to concrete problems in quantum sim-ulation. One potential route could involve splitting analgorithm up into small blocks, optimising each one andthen composing the optimised channels such that theoverall simulation is less noisy. This is particularly rel-evant to simulations of Trotterised quantum dynamicsand algorithms such as the quantum approximate opti-misation algorithm (QAOA), where layers of short-depth quantum circuits are repeated many times.Additionally, it would be instructive to implement theprotocol on other NISQ platforms. Whilst the frame-work presented here has been given in terms of digitalquantum gates, it is equally applicable to low level pulse-based quantum control and should be implementable onany quantum platform. Implementations on an NMRquantum computer would be of particular interest as theadvantages offered by the 0 − fidelity over the process fi-delity are more substantial. VI. ACKNOWLEDGMENTS
We are grateful to Rick Mukherjee for providing stim-ulating discussions. This work is supported by SamsungGRP grant, the UK Hub in Quantum Computing andSimulation, part of the UK National Quantum Technolo-gies Programme with funding from UKRI EPSRC grantEP/T001062/1 and the QuantERA ERA-NET Co-fundin Quantum Technologies implemented within the Euro-pean Union’s Horizon 2020 Programme. S.G. and F.S.are supported by studentships in the Quantum SystemsEngineering Skills and Training Hub at Imperial Col-lege London funded by EPSRC (EP/P510257/1). Weacknowledge the use of IBM Quantum services for thiswork. The views expressed are those of the authors, anddo not reflect the official policy or position of IBM or theIBM Quantum team. Numerical simulations were carriedout on Imperial HPC facilities [48].
VII. DATA AVAILABILITY
All of the data presented in this paper, along with thepython code used to generate this data and the figures,can be found in a public GitHub repository [49].
Appendix: Generation of Random Channels
The numerical analyses of the process and 0 − fidelitiesin Secs. II C and III C require the generation of randomunitary channels. In this appendix the procedure for ob-taining such channels is outlined.
1. Random Unitary Channels
In the numerical evaluations of the process and0 − fidelities in Figs. 1 and 2 and in the full trace esti-mations presented in Fig. 3, the channels Λ and Γ wereimplemented by target ( U t ) and comparison ( U c ) unitarymatrices as Λ( ρ ) = U t ρU † t and Γ( ρ ) = U c ρU † c . Randomtarget unitaries may be obtained by the exponentiationof random Hermitian matrices, which may themselves begenerated as H t = (cid:88) i ,i , ··· ,i n =0 α i i ··· i n n (cid:79) k =1 σ ( i k ) k , (A.1)where σ ( i k ) n are Pauli matrices acting on the k − th qubit(with σ (0) = ) and where the coefficients α i i ··· i n aresampled uniformly at random from the interval [ − , H t defined, the targetunitary is then given by U t = e − iH t .The coefficients for the target unitary used in the nu-merical simulations of the k − fidelities presented in Fig. 1are given in Table I. For the simulations of the 0 − fidelitiesshown in Fig. 2, the coefficients for the 2 and 3 qubit tar-get unitaries are given in Tables II and III whilst those forthe 5 qubit target unitary can be found in the GitHubrepository [49]. The coefficients for the target unitaryused in the full trace estimations presented in Fig. 3 aregiven in Table IV.For the comparison channels U c , one could generaterandom unitary matrices in the same way, however forbenchmarking purposes it is convenient to be able to con-trol the fidelities of the evaluated pairs of channels. Forthis reason, the comparison unitaries U c were obtained asunitary rotations of the target unitary U t generated bya random Hermitian matrix H r as U c = e − i(cid:15)H r U t e i(cid:15)H r ,where (cid:15) gives some control over the realised fidelities. ForFigs. 1 and 2, (cid:15) was varied from 0 to 1 to obtain eval-uations over a full range of fidelities, whilst for Fig. 3a single value of (cid:15) = 0 . H r used in the full traceestimations presented in Fig. 2 are given in Table V.
2. Random Quantum Circuits
The projective estimations of the process and0 − fidelities in Figs. 4 and 5 were performed using the cir- cuit in Fig. 9 with all 30 parameters sampled uniformly atrandom from the interval [0 , π ]. The parameters corre-spond to 10 U gates, each of which has three parameters θ, φ, λ which define the gate as U ( θ, φ, λ ) = (cid:18) cos θ/ − e iλ sin θ/ e iφ sin θ/ e i ( λ + φ ) cos θ/ (cid:19) . (A.2)The specific parameters used are shown in Table VI. Forthe experimental estimations presented in Fig. 5 the tar-get channel was obtained as the unitary representation ofthe circuit, and so any departure from the ideal fidelityarises from noise in the device.For the numerical simulations presented in Fig. 4 thetarget channel was obtained as the unitary representationof the circuit and the comparison channel was generatedby adding random coefficients sampled from the inter-val [ − . , .
4] to all of the U parameters, with the samepair of channels being used for all numbers of total ex-periments. The specific parameters used in this work are U U U U U U U U U U FIG. 9. Circuit for the generation of random channels inFigs. 4 and 5. Each of the U gates has three parameters θ, φ, λ which are randomly sampled from the interval [0 , π ]. given in Tables VII and VII. In these simulations variousnumbers of expectation values l and shots m were used;these experimental setups are given in Table IX. [1] N. Friis, O. Marty, C. Maier, C. Hempel, M. Holz¨apfel,P. Jurcevic, M. B. Plenio, M. Huber, C. Roos, R. Blatt, et al. , Observation of entangled states of a fully controlled20-qubit system, Phys. Rev. X , 021012 (2018).[2] B.-X. Wang, M.-J. Tao, Q. Ai, T. Xin, N. Lambert,D. Ruan, Y.-C. Cheng, F. Nori, F.-G. Deng, and G.-L. Long, Efficient quantum simulation of photosyntheticlight harvesting, npj Quantum Inf. , 1 (2018).[3] A. 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Kim, H r term α i i i H t term α i i i H t term α i i i H t term α i i i σ (0) σ (0) σ (0) σ (1) σ (0) σ (0) -0.70712 σ (2) σ (0) σ (0) -0.24903 σ (3) σ (0) σ (0) -0.55919 σ (0) σ (0) σ (1) σ (1) σ (0) σ (1) σ (2) σ (0) σ (1) σ (3) σ (0) σ (1) σ (0) σ (0) σ (2) σ (1) σ (0) σ (2) σ (2) σ (0) σ (2) σ (3) σ (0) σ (2) -0.10824 σ (0) σ (0) σ (3) σ (1) σ (0) σ (3) σ (2) σ (0) σ (3) -0.96490 σ (3) σ (0) σ (3) σ (0) σ (1) σ (0) σ (1) σ (1) σ (0) -0.89936 σ (2) σ (1) σ (0) -0.51887 σ (3) σ (1) σ (0) σ (0) σ (1) σ (1) σ (1) σ (1) σ (1) -0.97843 σ (2) σ (1) σ (1) σ (3) σ (1) σ (1) σ (0) σ (1) σ (2) -0.20607 σ (1) σ (1) σ (2) σ (2) σ (1) σ (2) -0.16243 σ (3) σ (1) σ (2) -0.66006 σ (0) σ (1) σ (3) σ (1) σ (1) σ (3) σ (2) σ (1) σ (3) σ (3) σ (1) σ (3) σ (0) σ (2) σ (0) -0.24559 σ (1) σ (2) σ (0) σ (2) σ (2) σ (0) -0.13876 σ (3) σ (2) σ (0) -0.90715 σ (0) σ (2) σ (1) -0.00605 σ (1) σ (2) σ (1) σ (2) σ (2) σ (1) σ (3) σ (2) σ (1) -0.66542 σ (0) σ (2) σ (2) σ (1) σ (2) σ (2) -0.29469 σ (2) σ (2) σ (2) σ (3) σ (2) σ (2) σ (0) σ (2) σ (3) σ (1) σ (2) σ (3) -0.32656 σ (2) σ (2) σ (3) σ (3) σ (2) σ (3) σ (0) σ (3) σ (0) σ (1) σ (3) σ (0) -0.49094 σ (2) σ (3) σ (0) -0.11147 σ (3) σ (3) σ (0) σ (0) σ (3) σ (1) -0.98569 σ (1) σ (3) σ (1) -0.01268 σ (2) σ (3) σ (1) -0.13383 σ (3) σ (3) σ (1) σ (0) σ (3) σ (2) σ (1) σ (3) σ (2) -0.70425 σ (2) σ (3) σ (2) σ (3) σ (3) σ (2) σ (0) σ (3) σ (3) -0.58409 σ (1) σ (3) σ (3) -0.86759 σ (2) σ (3) σ (3) σ (3) σ (3) σ (3) U t forthe numerical evaluation of the k − fidelities presented in Fig. 1. H t term α i i H t term α i i σ (0) σ (0) σ (2) σ (0) σ (0) σ (1) -0.28442 σ (2) σ (1) σ (0) σ (2) -0.36456 σ (2) σ (2) -0.12910 σ (0) σ (3) -0.11006 σ (2) σ (3) σ (1) σ (0) σ (3) σ (0) -0.47983 σ (1) σ (1) -0.70606 σ (3) σ (1) σ (1) σ (2) -0.66813 σ (3) σ (2) -0.59174 σ (1) σ (3) -0.60901 σ (3) σ (3) U t for the numerical evaluation of the two qubit 0 − fidelities presented in Fig. 2. H r term α i i i H t term α i i i H t term α i i i H t term α i i i σ (0) σ (0) σ (0) σ (1) σ (0) σ (0) -0.38200 σ (2) σ (0) σ (0) -0.84396 σ (3) σ (0) σ (0) σ (0) σ (0) σ (1) σ (1) σ (0) σ (1) -0.87782 σ (2) σ (0) σ (1) σ (3) σ (0) σ (1) σ (0) σ (0) σ (2) σ (1) σ (0) σ (2) -0.91405 σ (2) σ (0) σ (2) σ (3) σ (0) σ (2) σ (0) σ (0) σ (3) -0.12891 σ (1) σ (0) σ (3) σ (2) σ (0) σ (3) σ (3) σ (0) σ (3) -0.97166 σ (0) σ (1) σ (0) -0.66331 σ (1) σ (1) σ (0) σ (2) σ (1) σ (0) -0.78235 σ (3) σ (1) σ (0) σ (0) σ (1) σ (1) σ (1) σ (1) σ (1) -0.70895 σ (2) σ (1) σ (1) -0.63625 σ (3) σ (1) σ (1) -0.93680 σ (0) σ (1) σ (2) -0.00363 σ (1) σ (1) σ (2) -0.73675 σ (2) σ (1) σ (2) -0.86178 σ (3) σ (1) σ (2) -0.97715 σ (0) σ (1) σ (3) -0.68100 σ (1) σ (1) σ (3) -0.96935 σ (2) σ (1) σ (3) σ (3) σ (1) σ (3) σ (0) σ (2) σ (0) -0.32367 σ (1) σ (2) σ (0) σ (2) σ (2) σ (0) -0.78411 σ (3) σ (2) σ (0) σ (0) σ (2) σ (1) -0.25096 σ (1) σ (2) σ (1) -0.98511 σ (2) σ (2) σ (1) σ (3) σ (2) σ (1) -0.02550 σ (0) σ (2) σ (2) σ (1) σ (2) σ (2) -0.11461 σ (2) σ (2) σ (2) σ (3) σ (2) σ (2) σ (0) σ (2) σ (3) σ (1) σ (2) σ (3) -0.22837 σ (2) σ (2) σ (3) σ (3) σ (2) σ (3) -0.02733 σ (0) σ (3) σ (0) σ (1) σ (3) σ (0) σ (2) σ (3) σ (0) σ (3) σ (3) σ (0) σ (0) σ (3) σ (1) σ (1) σ (3) σ (1) σ (2) σ (3) σ (1) -0.09857 σ (3) σ (3) σ (1) σ (0) σ (3) σ (2) σ (1) σ (3) σ (2) σ (2) σ (3) σ (2) σ (3) σ (3) σ (2) σ (0) σ (3) σ (3) σ (1) σ (3) σ (3) σ (2) σ (3) σ (3) -0.14916 σ (3) σ (3) σ (3) U t for the numerical evaluation of the three qubit 0 − fidelities presented in Fig. 2. H t term α i i i H t term α i i i H t term α i i i H t term α i i i σ (0) σ (0) σ (0) -0.12226 σ (1) σ (0) σ (0) σ (2) σ (0) σ (0) σ (3) σ (0) σ (0) -0.34315 σ (0) σ (0) σ (1) -0.54535 σ (1) σ (0) σ (1) -0.24266 σ (2) σ (0) σ (1) σ (3) σ (0) σ (1) -0.60957 σ (0) σ (0) σ (2) σ (1) σ (0) σ (2) σ (2) σ (0) σ (2) σ (3) σ (0) σ (2) σ (0) σ (0) σ (3) -0.83056 σ (1) σ (0) σ (3) -0.92854 σ (2) σ (0) σ (3) σ (3) σ (0) σ (3) -0.09843 σ (0) σ (1) σ (0) -0.44200 σ (1) σ (1) σ (0) -0.22618 σ (2) σ (1) σ (0) σ (3) σ (1) σ (0) -0.99722 σ (0) σ (1) σ (1) -0.65783 σ (1) σ (1) σ (1) σ (2) σ (1) σ (1) -0.93251 σ (3) σ (1) σ (1) -0.61818 σ (0) σ (1) σ (2) -0.72367 σ (1) σ (1) σ (2) σ (2) σ (1) σ (2) σ (3) σ (1) σ (2) -0.91497 σ (0) σ (1) σ (3) -0.06722 σ (1) σ (1) σ (3) σ (2) σ (1) σ (3) -0.26926 σ (3) σ (1) σ (3) σ (0) σ (2) σ (0) -0.75042 σ (1) σ (2) σ (0) -0.78464 σ (2) σ (2) σ (0) -0.30934 σ (3) σ (2) σ (0) σ (0) σ (2) σ (1) -0.75602 σ (1) σ (2) σ (1) σ (2) σ (2) σ (1) σ (3) σ (2) σ (1) σ (0) σ (2) σ (2) σ (1) σ (2) σ (2) σ (2) σ (2) σ (2) σ (3) σ (2) σ (2) -0.22242 σ (0) σ (2) σ (3) -0.99924 σ (1) σ (2) σ (3) -0.80395 σ (2) σ (2) σ (3) σ (3) σ (2) σ (3) σ (0) σ (3) σ (0) σ (1) σ (3) σ (0) σ (2) σ (3) σ (0) σ (3) σ (3) σ (0) -0.23753 σ (0) σ (3) σ (1) -0.95405 σ (1) σ (3) σ (1) -0.90343 σ (2) σ (3) σ (1) -0.93316 σ (3) σ (3) σ (1) σ (0) σ (3) σ (2) -0.42508 σ (1) σ (3) σ (2) -0.66899 σ (2) σ (3) σ (2) -0.77155 σ (3) σ (3) σ (2) σ (0) σ (3) σ (3) -0.31457 σ (1) σ (3) σ (3) -0.19482 σ (2) σ (3) σ (3) -0.92350 σ (3) σ (3) σ (3) -0.54485TABLE IV. Table of coefficients in the random Hermitian matrix (Eq. (A.1)) used to generate the random target unitary U t for the numerical evaluation of the full trace process and 0 − fidelity estimations presented in Fig. 3. H r term α i i i H r term α i i i H r term α i i i H r term α i i i σ (0) σ (0) σ (0) σ (1) σ (0) σ (0) σ (2) σ (0) σ (0) -0.16832 σ (3) σ (0) σ (0) -0.94659 σ (0) σ (0) σ (1) σ (1) σ (0) σ (1) σ (2) σ (0) σ (1) σ (3) σ (0) σ (1) σ (0) σ (0) σ (2) -0.17869 σ (1) σ (0) σ (2) σ (2) σ (0) σ (2) -0.01940 σ (3) σ (0) σ (2) σ (0) σ (0) σ (3) -0.30403 σ (1) σ (0) σ (3) σ (2) σ (0) σ (3) -0.07383 σ (3) σ (0) σ (3) -0.55021 σ (0) σ (1) σ (0) -0.79330 σ (1) σ (1) σ (0) -0.27526 σ (2) σ (1) σ (0) σ (3) σ (1) σ (0) σ (0) σ (1) σ (1) -0.85750 σ (1) σ (1) σ (1) -0.30359 σ (2) σ (1) σ (1) -0.02089 σ (3) σ (1) σ (1) σ (0) σ (1) σ (2) σ (1) σ (1) σ (2) -0.47171 σ (2) σ (1) σ (2) σ (3) σ (1) σ (2) -0.42033 σ (0) σ (1) σ (3) σ (1) σ (1) σ (3) -0.99054 σ (2) σ (1) σ (3) σ (3) σ (1) σ (3) -0.00098 σ (0) σ (2) σ (0) -0.85538 σ (1) σ (2) σ (0) σ (2) σ (2) σ (0) σ (3) σ (2) σ (0) σ (0) σ (2) σ (1) -0.94635 σ (1) σ (2) σ (1) σ (2) σ (2) σ (1) σ (3) σ (2) σ (1) σ (0) σ (2) σ (2) σ (1) σ (2) σ (2) -0.46577 σ (2) σ (2) σ (2) -0.44707 σ (3) σ (2) σ (2) -0.73448 σ (0) σ (2) σ (3) -0.77559 σ (1) σ (2) σ (3) -0.36554 σ (2) σ (2) σ (3) -0.67578 σ (3) σ (2) σ (3) σ (0) σ (3) σ (0) -0.59413 σ (1) σ (3) σ (0) -0.42969 σ (2) σ (3) σ (0) σ (3) σ (3) σ (0) -0.98854 σ (0) σ (3) σ (1) σ (1) σ (3) σ (1) σ (2) σ (3) σ (1) -0.95976 σ (3) σ (3) σ (1) σ (0) σ (3) σ (2) σ (1) σ (3) σ (2) σ (2) σ (3) σ (2) σ (3) σ (3) σ (2) -0.38816 σ (0) σ (3) σ (3) σ (1) σ (3) σ (3) σ (2) σ (3) σ (3) -0.35423 σ (3) σ (3) σ (3) -0.18427TABLE V. Table of coefficients in the random Hermitian matrix (Eq. (A.1)) used to generate the random comparison unitary e − iH r for the numerical evaluation of the full trace process and 0 − fidelity estimations presented in Fig. 3. U gate θ φ λU (1)3 U (2)3 U (3)3 U (4)3 U (5)3 U (6)3 U (7)3 U (8)3 U (9)3 U (10)3 U parameters used in the circuit in Fig. 9 for the generation of a random benchmarking channel. Thechannel was implemented on the IBM Q Toronto quantum device, with all deviations from ideality arising from noise in themachine. U gate θ φ λU (1)3 U (2)3 U (3)3 U (4)3 U (5)3 U (6)3 U (7)3 U (8)3 U (9)3 U (10)3 U parameters used in the circuit in Fig. 9 for the generation of a target unitary U t . U gate θ φ λU (1)3 U (2)3 U (3)3 U (4)3 U (5)3 U (6)3 U (7)3 U (8)3 U (9)3 U (10)3 U parameters used in the circuit in Fig. 9 for the generation of the comparison unitary.0 − fidelity Process fidelityTotal Experiments lm Expectation values l Shots m Expectation values l Unique experiments dl Shots m
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