Digital Quantum Simulation of Non-Equilibrium Quantum Many-Body Systems
DDigital Quantum Simulation of Non-Equilibrium Quantum Many-Body Systems
Benedikt Fauseweh ∗ and Jian-Xin Zhu
1, 2, † Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: September 17, 2020)Digital quantum simulations (DQS) is a driving force behind the development of universal quan-tum computers. DQS uses the capabilities of quantum computers, such as superposition and en-tanglement, to determine the dynamics of quantum systems, which are beyond the computabilityof modern classical computers. A notoriously challenging task in this field is the description of non-equilibrium dynamics in quantum many-body systems, because it involves a macroscopic numberof excitations above the ground state, and defies the methods and principles of equilibrium sys-tems. Here we use the IBM quantum computers to simulate the non-equilibrium dynamics of fewspin and fermionic systems. We explicitly include external perturbations, such as pulsed magneticfields, to model excitation mechanisms in more realistic situations. Our results reveal, that witha combination of error mitigation, noise extrapolation and optimized initial state preparation, onecan tackle the most important drawbacks of modern quantum devices. Our results culminate inthe first experimental observation of the dynamical breakdown of the Fermi surface due to stronginteractions on a digital quantum computer. The systems we simulate demonstrate the potentialfor large scale quantum simulations of light-matter interactions in the near future.
INTRODUCTION
Discovering ways to control the dynamical aspects ofquantum materials is an important research frontier inmodern solid state physics. Advances in non-linear op-tics and ultrafast spectroscopy allowed to induce non-equilibrium states with new properties on picosecondtime scales. In recent years many fascinating phenom-ena have been observed with this approach, such as lightinduced superconductivity [1, 2] or ultra fast switchingof topological properties [3]. These phenomena can leadto desirable electronic and magnetic properties for futuredevices with potentially groundbreaking applications.Theoretical understanding or even prediction of theseeffects is still scarce, due to computational challenges.The main issue comes from the many interacting particlesthat participate in such non-equilibrium phenomena, asthe classical computational effort to simulate a system ofquantum particles scales exponentially in their number N . This is called the curse of dimensionality.A possible solution to this problem was proposed byFeynman in 1982 in the form of digitial quantum simu-lation (DQS) [4]. It is based on the idea that a quantumsystem is best simulated by another quantum system.The main ingredient for DQS is a universal quantumcomputer. A quantum computer uses the effects of quan-tum mechanics to execute algorithms, which are substan-tially faster than their classical counterparts. Quantumcomplexity theory describes the advantages of these algo-rithms by providing lower bounds for the time complex-ity when compared to a classical Turing machine. Withquantum complexity theory it is possible to prove thatthe simulation of quantum systems scales only polynomi-ally on a general purpose quantum computer [5].This theoretical superiority has the price of a number of technical challenges [6], the biggest one being deco-herence of the quantum information, i.e. the decay ofthe information encoded in a quantum register. In re-cent years several different approaches were successful inbuilding small scale quantum computers. While first im-plementations were based on nuclear magnetic resonancein liquids [7–9], recent implementations have been basedon superconducting circuits [10] and trapped ions [11].However, with the increase of qubits in these quantumdevices (up to 72 qubits nowadays), the noise and er-ror rates prohibit high fidelity use of all qubits for faulttolerant quantum computation [12–14]. Therefore, re-cent research in DQS has been focused on evaluating theperformance of these Noisy Intermediate-Scale Quantum(NISQ) devices for the purpose of time-dependent quan-tum simulation [15–24]. These studies showed that, inits pure form, DQS is still in its infancy on NISQ devicesdue to the low fidelity and decoherence of the qubits, sug-gesting that dealing with errors has the highest priority.Here we tackle this problem by incorporating a seriesof state-of-the-art methods in order to improve the qual-ity of DQS. The performance of the approach is demon-strated on the IBM quantum computers by simulatingnon-equilibrium excitations in various systems. We per-form three classes of experiments with increasing com-plexity to evaluate how different aspects of the simula-tion, such as error mitigation, zero noise extrapolationand optimal state preparation, can affect the circuit andquality of results on NISQ devices. RESULTS
Non-equilibrium DQS consists of three fundamentalsteps. First the initial state is prepared, e.g. an incom-ing scattering state or the ground state of a Hamilto- a r X i v : . [ qu a n t - ph ] S e p a)b) c) FIG. 1.
Quantum simulation of the Rabi resonance experiment. a)
Graphical representation of the spin movement onthe bloch sphere. The initial state depends on the angle α which rotates the spin between the states | (cid:105) = | ↑(cid:105) and | (cid:105) = | ↓(cid:105) .The magnetic field (cid:126)H ( t ) is constant in magnitude and precesses around the z axis with angle Θ and angular velocity ω . b) Quantum circuit to simulate the Rabi experiment on a digitial quantum computer. Initially the state is rotated by an R y gateby the angle α . The unitary time evolution is discretized t → t + ∆ t in order to evolve the spin. To reach a time t = N ∆ t werequire N single qubit gates U ( t, ∆ t ). Finally the z component of the spin is measured. c) Results from the quantum simulationfor the parameters H = 1 , Θ = 2 , ω = 1 , α = 2 π/
3. Comparison between exact results for the probability of measuring eitherup or down spin, the classically simulated discretized time evolution, the bare experimental results and the error mitigatedexperimental results. nian. Second the state is time evolved under the actionof time-dependent Hamiltonian. Finally observables aremeasured after the time evolution. Each of these stepshas its own sources of errors, such as imperfect initializa-tion or readout errors. We start by evaluating the sin-gle qubit performance of the IBM quantum computerswith respect to the simulation of non-equilibrium sys-tems. Specifically we will simulate the spin S = 1 / (cid:126)H ( t ), H = (cid:126)S · (cid:126)H ( t ) . (1)This model was first discussed by Rabi in 1937 [25] anddescribes the cyclic behaviour of a two-level quantum sys-tem under periodic external perturbation. We identifythe computational basis with the spin basis, using theFeynman-Vernon-Hellwarth picture [26]. As initial con-dition we start with a fully polarized spin which is sub-sequently rotated by an angle α with a R y gate. The ex-ternal magnetic field is rotating in the x - y plane and hasa fixed angle Θ with respect to the z axis. An overviewof the system is shown in Fig. 1 a). The quantum circuitfor the simulation of the spin dynamics is shown in Fig.1 b). After the initial rotation we discretize the time do-main and apply the unitary time evolution U ( t, ∆ t ) at each time step. Each time evolution can be mapped ontothe universal U z component is measured. In Fig. 1 c) we investigate theresults for a representative set of parameters. While thebare experimental results (blue) follow the exact resultsqualitatively, there is a systematic offset. This offset canbe traced back to the readout error of the device. Ifwe assume that this readout error is independent of thegates applied before readout, we can perform a calibra-tion measurement of the single qubit in order to applyreadout error mitigation, see Methods section. This ap-proach allows us to significantly reduce the error of ourquantum simulation (red). Only in the long time limitwe see that decoherence and gate imperfections gradu-ally reduce the accuracy. Note that the error that stemspurely from time discretization is negligible (black).Next we investigate the non-equilibrium excitation of a)b)c) d)e)f) FIG. 2.
Nonequilibrium quantum simulation of spin models subject to magnetic field pulses.
Only the non-vanishing spin-spin coupling is shown. a) Time dependence of the magnetic field pulse used in b) - d) . The circular polarizedpulse is parameterized as H x ( t ) = h exp (cid:0) − ( t − t ) / (2 τ ) (cid:1) cos( ω ( t − t )) and H y ( t ) = h exp (cid:0) − ( t − t ) / (2 τ ) (cid:1) sin( ω ( t − t ))with parameters h = 2 , ω = 1 , τ = 0 . , t = 2. b) Simulation of a spin dimer in a time-dependent magnetic field. Comparisonbetween the exact time evolution of the states, the bare experimental data, the readout error mitigated data and the classicallycomputed trotterized simulation. c) Time evolution of the total magnetization S z ( t ) = (cid:104) ψ ( t ) | S z + S z | ψ ( t ) (cid:105) derived from b) . d) Simulation of a spin plaquette. Displayed is the staggered magnetization S AF = (cid:80) i =1 ( − i S iz . Comparison between thebare experimental results, the readout error mitigated results and the combination of readout error mitigation and zero noiseextrapolation. e) Time and site resolved magnetization for an edge driven spin chain with 8 spins. Comparison between the exactresults, experimental results after readout error mitigation and the combination of error mitigation and zero noise extrapolation. f )
Pulse used for the edge drive in e) . The pulse is linear polarized and parameterized as H x ( t ) = h exp (cid:0) − ( t − t ) / (2 τ ) (cid:1) and H y ( t ) = 0 with parameters h = π/ , τ = 1 , t = 1 . coupled spin systems. Specifically we investigate pulsedspin dimers and spin plaquettes and an edge driven spinchain with eight spins. The general form of the Hamilto-nian we investigate reads H = (cid:88) (cid:104) i,j (cid:105) J ⊥ (cid:0) S xi S xj + S yi S yj (cid:1) + J z S zi S zj + (cid:88) i (cid:126)H i ( t ) (cid:126)S i , (2)where (cid:126)H i ( t ) is a site- and time-dependent magnetic field.We use 2nd order Trotterization to discretize the timeevolution of the Hamiltonian, see Methods section. Theresults are shown in Fig. 2. We observe that the reach-able time scales before large errors set in have been re- duced almost by an order of magnitude as compared tothe single spin case. This effect can be traced back to thetwo qubit gate errors, specifically the CNOT gate errors,which are typically in the range of 0 .
5% up to 3%, de-pending on the device and the qubits used, see Methodssection. Readout error mitigation still significantly re-duces the error when compared to the bare experimentalresults and it has a large effect on global quantities, suchas the total magnetization, Fig. 2 c), or the staggeredmagnetization, Fig. 2 d), while the probabilities of thespecific states, such as in the spin dimer in Fig. 2 b),show larger errors, even after readout error mitigation ismade. a)b) c)d)
FIG. 3.
Quantum simulation of an interaction quench in a fermion system.
All experimental data are obtained bycombination of optimal state preparation, readout and symmetry error mitigation, zero noise extrapolation and full quantumstate tomography after each time step. a) Time and momentum dependence of the fermionic distribution function on a periodicfour site chain after an interaction quench to U final = 2 at t = 0. b) Time evolution of the filling factor. Comparison betweenthe exact filling, the weak quench U final = 1 and the strong quench U final = 2. c) Time evolution of the jump in the Fermidistribution. Comparison between the experimental results for weak and strong quenches and classical simulations. For theclassical simulation the state after initial preparation at t = 0 was obtained by full state tomography from the quantumcomputer. It was subsequently evolved by the Schroedinger equation, orange and blue line, and by Trotterization, black dashedline, on a classical computer. d) Entanglement entropy of a bipartition of the system. The classical simulations where obtainedwith the same method as in c) . To reduce the additional noise coming from the CNOTgates in the DQS circuit we apply zero noise extrapola-tion [27, 28]. The basic idea of this approach is to boostthe errors from the two-qubit gates artificially, but ina controlled fashion, in order to map them to the zeronoise case. To apply this approach, it is important thatthe noise itself is time-invariant, i.e. it is invariant un-der time rescaling. For the superconducting qubits ofthe IBM quantum devices this was indeed doable [29].The extrapolation is implemented together with readouterror mitigation for the spin plaquette in Fig. 2 d) andadditionally with symmetry error mitigation for a locallydriven spin chain in Fig. 2 e). Symmetry error mitigationuses the fact, that there is conserved mirror symmetrywith respect to the center of the chain, even during exci-tation with an external magnetic field. This mirror sym-metry is not necessarily conserved in the quantum circuit, due to gate imperfections. We therefore symmetrize theresults from the quantum computer to correct for errorsviolating this symmetry.Individually, these improvements are insufficient to sig-nificantly enhance the quality of the experimental results.However, a synergy effect from the combination can leadto quantitative agreement between the experimental re-sults and the exact time evolution.So far we have studied the non-equilibrium dynam-ics of quantum systems starting from a state which isfactorizable. In real materials this situation is typicallydifferent. Even at zero temperature quantum fluctua-tions lead to ground states which are highly entangled.One of the open questions in this context is how suchclosed many-body system relax after excitation. Here weaddress this questions for a chain of interacting spinlessfermions. This model is equivalent to the XXZ chain, H = (cid:88) i ∈ Z J ⊥ (cid:0) S xi · S xi +1 + S yi · S yi +1 (cid:1) + U ( t ) S zi · S zi +1 , (3)by virtue of the Jordan-Wigner transformation [30] andit is integrable with the Bethe ansatz [31]. We use a qubitplaquette to simulate a four -site chain with a periodicboundary condition. The system is in its non-interactingground state for U ( t = 0) = 0 and then a sudden quan-tum quench of the interaction is performed. Such non-adiabatic interaction quenches in Fermi systems are asubject of ongoing research [32–37].To initialize the system in its highly entangled groundstate we compute the state amplitudes classically andthen apply optimized state preparation [38, 39] to re-duce the number of noisy CNOT gates. The resultingcircuit is given in the Methods section. We apply read-out and symmetry error mitigation as well as zero noiseextrapolation. After initialization the ground state hasa fidelity of 94% and is pure within statistical error. Wethen quench the interaction strength in the Hamiltonianand time evolve the state using Trotterization. We com-pare the effect of two quench strengths U final = 1 and U final = 2. To compute the time evolution of the mo-mentum distribution we use state tomography after thetime evolution and then measure the occupation for eachmomentum space point. The results are given in Fig. 3.We observe that although we only have four different mo-mentum space points, the breakdown of the Fermi surfacedue to interactions is clearly identifiable. Note that theoverall occupation seems to be stable with respect to thestate preparation and time evolution, as the error staysbelow 4%(Fig. 3 b)). We trace this observation to thefact that, even in the maximally mixed state, i.e. withoutany coherence in the system, the total filling is exactly 2.In order to compare the results to classical simulationswe compute the time evolution using the density matrixobtained from tomography immediately after initial statepreparation to assess the errors coming purely from timeevolution (blue and orange curves in Fig. 3 c) and d)).The speed of the breakdown and the dependence on theinteraction quench is compared in Fig. 3 c). There is agood agreement between these classical simulations andthe quantum computer experiments. The precision ofthe simulation allows us to clearly distinguish betweenthe different strengths of the quantum quench. We alsocompare the time evolution of the entanglement entropyfor a bipartition of the system in Fig. 3 d).We noticethat already at t = 0 the state is highly entangled due tothe fermionic nature of the ground state. The time evo-lution of the entanglement entropy is only qualitativelycaptured by the quantum computer, with larger errorswhen compared to the jump of the Fermi surface. CONCLUSION
In this paper we have demonstrated the capabilitiesof modern NISQ devices with respect to simulating non-equilibrium dynamics in quantum system. Using a seriesof optimizations to improve the accuracy of the simula-tion circuits, we could significantly outperform DQS inits bare form. While single spin dynamics can be easilycaptured quantitatively with simple readout error mitiga-tion, multi-spin systems are subjected to much strongererrors, due to the low two-qubit gate fidelities, reduc-ing simulatable time scales when compared to the singlespin case. We therefore applied more refined strategies,such as symmetrization and zero noise extrapolation tosimulate pulse driven spin dimers, plaquettes and chains.Finally we focused on highly entangled initial states andcombined all previous mitigation techniques with optimalinitial state preparation to simulate interaction quenchdynamics in small scale Fermi systems.Our work highlights the importance of sophisticatedmethods in order to simulate non-equilibrium dynamicson modern quantum processors. So far only small timescales and system sizes can be reached, due to noise andgate imperfections. Recent improvements of the IBMquantum devices [40], such as shorter two-qubit gates anddynamic decoupling sequences, can be combined with ourapproach as well. With further technological improve-ments we can reach system sizes which are not tractableby classical computers, making DQS a possible candidatefor the demonstration of quantum supremacy. NISQ de-vices available today have already reached the number ofrequired qubits ( ∼ METHODS
IBM Quantum Experience devices and qubits.
To com-pute the time evolution in the main paper we used threedifferent devices from the IBM Quantum Experience. Forthe experiments shown in Figs. 1 and 2 we used the 27qubit devices ibmq toronto and ibmq montreal. Theyhave an identical architecture, which is shown in Fig. 4a). For the calculations on the fermionic system we usedthe 5 qubit device ibmq bogota, shown in Fig. 4 b). Notethat these devices are recalibrated on a daily basis, whichmay change the characteristics of the gate error rates aswell as the decoherence times. We have summarized thecalibration of the qubits used in the experiments in theTables I, II, III, IV and V. All results from quantumdevices were averaged over 8192 samples, such that thestatistical error is negligible.
Readout error mitigation.
To reduce the error comingfrom readout, we applied a readout error mitigation tech-nique [41, 42]. Two primary assumptions are necessaryfor this approach: 1) The error coming from readout isdue to classical noise and 2) that noise is independentof the quantum gates applied to the system beforehand.In recent study it was shown that classical noise is in-deed the dominant noise on the IBM quantum devices[43]. We thus executed a calibration experiment beforeeach time evolution experiment in order to characterizethe device for each of the 2 N basis states. Although thisapproach scales exponentially in the system size, this canbe overcome by tensored error mitigation, assuming fur-ther that the error by noise is local and correlates onlya subset of qubits. After the calibration experiment, we arranged the results of each calibration experiment in a2 N × N matrix Λ, Λ ij = p ( i, j ) (4)where p ( i, j ) is the probability of preparing state i andmeasuring state j . If the single gate errors are smallcompared to the readout noise this matrix perfectly mapsthe ideal results to the experimental results by means ofa classical post processing of the statistics p exp = Λ p ideal . (5)Thus to obtain the ideal results we applied the inverseof Λ onto the experimental results of our time evolution.Note that the inverse might not be well defined, due tostrong noise. In this case the MoorePenrose pseudoin-verse was applied to obtain a least-square solution. Time evolution.
To time evolve the quantum statesafter initialization, we used symmetric Trotterization todecompose the time evolution operator. U ( t, ∆ t ) = T exp − i t +∆ t (cid:90) t H ( τ )d τ = exp (cid:20) − i ∆ t H A (cid:18) t + ∆ t (cid:19)(cid:21) exp (cid:20) − i ∆ tH B (cid:18) t + ∆ t (cid:19)(cid:21) exp (cid:20) − i ∆ t H A (cid:18) t + ∆ t (cid:19)(cid:21) + O (∆ t ) , (6)where H A and H B are two in general non-commutingparts of the Hamiltonian. Note that this approach canbe generalized to more non-comuting parts as well [44].In some cases we also used a lower order formula if themain source of error comes from gate imperfections andnot from Trotterization. In the Hamiltonians we are sim-ulating the interactions are strongly localized. Therefore,we need only one and two-qubit gates to fully implementthe necessary time evolution gates. For details on theimplementation of these gates we refer to the literature[15]. Zero noise extrapolation.
In order to perform zeronoise extrapolation we concentrate on the two qubit gatesand assume that the single qubit gates have negligibleerror rates in comparison. This is supported by the ob-servation that the two qubit error rates are typically oneorder of magnitude larger than the single qubit gates onthe IBM quantum devices, see above. Zero noise ex-trapolation is based on the idea of Richardson’s deferredapproach to the limit. Suppose we want to measure anexpectation value of an observable E . Then this expec-tation value can be expanded as a power series in a noise parameter λ , E ( λ ) = E + ∞ (cid:88) k =1 a k λ k , (7)where E is the zero noise limit. If we can ob-tain several expectation values at different noise values E ( λ ) , E ( λ ) , . . . it is possible to cancel out the leadingcontribution in the power series to obtain a better esti-mate of the zero noise term. To obtain these differentnoise values it is possible to scale the whole quantum cir-cuit by a factor c . In Ref. [29] this approach was appliedto the single as well as to the two qubit gates, by rescal-ing the duration of the microwave pulses. This rescalingrequires a precise recalibration of the two qubit gates dueto non-linearities in the amplitude dependence. Here weuse a simpler scheme by concentrating only on the CNOTgates and fix the stretching factor to c = 3. Specificallywe set up the quantum circuits for c = 1 and then re-place every CNOT gate but three times the same CNOTgate. This approach allows us to boost the noise from theCNOT gate and at the same time perform the same com-putation, as in theory three repeated CNOTs are equiv-alent to a single CNOT. We then use the readout of theextended circuit to perform a linear extrapolation of theexpectation value, assuming that the single qubit gateshave no error and do not contribute to the noise. Fermionic model.
In order to simulate the breakdown of the Fermi surface in Fig. 3 we are simulating the XXZmodel on a plaquette as given in Eq. (3). Using the Jor-dan Wigner transformation [30] we obtain an equivalentfermionic model, H LL = J ⊥ (cid:88) i =1 (cid:16) c † i c i +1 + h.c. (cid:17) + J ⊥ − N − (cid:16) c † c + h.c. (cid:17) + U (cid:88) i ∈ Z (cid:18) c † i c i − (cid:19) (cid:18) c † i +1 c i +1 − (cid:19) , (8)with U = J ⊥ ∆ and N the total number of fermions.Note that we use periodic boundary conditions for thespin model. In this case the ground state for U = 0 is inthe sector with N = 2 fermions. Therefore the fermionicmodel has anti-periodic boundary conditions, which re-sults in the k values − π/ , − π/ , π/ , π/
2. Even af-ter the quantum quench the fermionic Hamiltonian com-mutes with the total particle number operator and there-fore we will stay in the N = 2 sector during the timeevolution, also seen in Fig. 3 b). Optimal state preparation.
For the simulation of theinteraction quench in Fig. 3 we require a highly entan-gled initial state. Although in theory every state can beconstructed on a universal quantum computer [45], re-cent progress was on minimizing the number of requiredCNOT gates in order to increase the fidelity of the statepreparation [39, 46]. Here we use the universal Q com-piler [38] to compute an optimized circuit for the prepa-ration of the fermionic ground state. The result in QASMcode is given in Listing 1 and shown in Fig. 5.
State tomography and entanglement entropy.
In orderto compute the momentum distribution and the entan-glement entropy in Fig. 3, we used full state tomographyto deduce the state after each time step. This requires3 N circuits to measure the state in the X , Y and Z ba-sis. Due to noise during the quantum computation, thestate is not necessarily a pure state. We used a fitter im-plemented in the QISKIT package [41] to compute thedensity matrix ρ ( t ) based on a method introduced inRef. 47. We then used this density matrix to deducethe momentum distribution n k ( t ) = 1 N s (cid:88) i,j (cid:104) c † i c j (cid:105) ( t ) e ik ( r i − r j ) , (9)where N s = 4 is the number of sites and r i is the positionof site i . The expectation value (cid:104) . . . (cid:105) is with respect tothe density matrix ρ ( t ). The Von Neumann entanglemententropy is computed as S ( ρ A )( t ) = − Tr [ ρ A ( t ) log ρ A ( t )] , (10)where ρ A ( t ) = Tr B ( ρ ( t )) is the reduced density matrix ofsubsystem A. Here we used a bipartition of the system to define the subsystems A and B. Although this way ofcomputing the entanglement entropy does not scale tolarge system sizes, there are demonstrations on how tomeasure fermionic entanglement on quantum computersusing an alternate approach [48]. ACKNOWLEDGMENTS
We thank Christopher Lane, Zhao Huang, and AndrewSornborger for useful discussions. This work was carriedout under the auspices of the U.S. Department of Energy(DOE) National Nuclear Security Administration underContract No. 89233218CNA000001, and was supportedby the LANL LDRD Program. This research used re-sources provided by the LANL Institutional ComputingProgram. a) b)
FIG. 4.
Architecture of the IBM quantum devices used in the paper. a) Architecture of the ibmq montreal andibmq toronto devices. b) Architecture of the ibmq bogota device. Lines show the possible CNOT gates on these devices.Qubit t [ µ s] t [ µ s] U3 error rate readout error rate14 135 221 3 . · − . · − TABLE I. Device calibration for ibmq toronto device used in Fig. 1 a). Experiment executed on July 15th 2020.Qubit t [ µs ] t [ µs ] U3 error rate CNOT error rate readout error rate19 93 75 5 . · − →
20: 6 . · − . · −
20 88 130 4 . · − →
19: 6 . · − . · − TABLE II. Device calibration for ibmq montreal device used in Fig. 2 b) and c). Experiment executed on August 1st 2020.Qubit t [ µs ] t [ µs ] U3 error rate CNOT error rate readout error rate8 72 135 1 . · − →
11: 6 . · − . · −
11 94 142 4 . · − →
8: 6 . · − , 11 →
14: 1 . · − . · −
14 89 134 4 . · − →
11: 1 . · − , 14 →
16: 6 . · − . · −
16 85 124 4 . · − →
14: 6 . · − . · − TABLE III. Device calibration for ibmq toronto device used in Fig. 2 d). Experiment executed on September 4th 2020.Qubit t [ µs ] t [ µs ] U3 error rate CNOT error rate readout error rate4 126 141 4 . · − →
1: 1 . · − . · − . · − →
4: 1 . · − , 1 →
2: 9 . · − . · − . · − →
1: 9 . · − , 2 →
3: 1 . · − . · − . · − →
2: 1 . · − , 3 →
5: 6 . · − . · − . · − →
3: 6 . · − , 5 →
8: 6 . · − . · − . · − →
5: 6 . · − , 8 →
11: 7 . · − . · −
11 116 47 7 . · − →
8: 7 . · − , 11 →
14: 7 . · − . · −
14 138 134 9 . · − →
11: 7 . · − . · − TABLE IV. Device calibration for ibmq montreal device used in Fig. 2 e). Experiment executed on September 4th 2020.Qubit t [ µs ] t [ µs ] U3 error rate CNOT error rate readout error rate1 163 146 3 . · − →
2: 6 . · − . · − . · − →
1: 6 . · − , 2 →
3: 9 . · − . · − . · − →
2: 9 . · − , 3 →
4: 7 . · − . · − . · − →
3: 7 . · − . · − TABLE V. Device calibration for ibmq bogota device used in Fig. 3. Experiment executed on September 4th 2020.
FIG. 5.
Circuit for optimized state preparation.
Ground state preparation used in Fig. 3 of the main text.Listing 1. QASM code for optimal state preparationi n c l u d e ” q e l i b 1 . i n c ” ;qreg q [ 4 ] ;u3 ( 6 . 8 0 6 7 8 4 0 8 2 7 7 8 , 0 , 0) q [ 0 ] ;u3 ( 1 1 . 5 1 9 1 7 3 0 6 3 1 6 2 , − p i /2 , p i /2) q [ 1 ] ;cx q [ 0 ] , q [ 1 ] ;u3 ( 1 1 . 9 5 0 8 9 0 9 0 5 6 8 9 , 0 , 0) q [ 0 ] ;u3 ( 1 0 . 3 8 0 0 9 4 5 7 8 8 9 4 , 0 , 0) q [ 1 ] ;u1 ( 7 . 8 5 3 9 8 1 6 3 3 9 7 4 ) q [ 0 ] ;u3 ( 1 0 . 9 9 5 5 7 4 2 8 7 5 6 4 , − p i /2 , p i /2) q [ 2 ] ;cx q [ 0 ] , q [ 2 ] ;u3 ( 8 . 6 3 9 3 7 9 7 9 7 3 7 2 , − p i /2 , p i /2) q [ 1 ] ;u3 ( 1 0 . 9 9 5 5 7 4 2 8 7 5 6 4 , − p i /2 , p i /2) q [ 3 ] ;cx q [ 1 ] , q [ 3 ] ;u3 ( 1 1 . 7 8 0 9 7 2 4 5 0 9 6 2 , 0 , 0) q [ 3 ] ;u3 ( 1 1 . 7 8 0 9 7 2 4 5 0 9 6 2 , 0 , 0) q [ 2 ] ;u3 ( 9 . 4 2 4 7 7 7 9 6 0 7 6 9 , − p i /2 , p i /2) q [ 2 ] ;cx q [ 3 ] , q [ 2 ] ;u3 ( 1 1 . 7 8 0 9 7 2 4 5 0 9 6 2 , − p i /2 , p i /2) q [ 3 ] ;u1 ( 1 0 . 9 9 5 5 7 4 2 8 7 5 6 4 ) q [ 3 ] ;u1 ( 1 1 . 7 8 0 9 7 2 4 5 0 9 6 2 ) q [ 2 ] ;cx q [ 3 ] , q [ 2 ] ;u3 ( 1 0 . 9 9 5 5 7 4 2 8 7 5 6 4 , 0 , 0) q [ 3 ] ;u1 ( 8 . 6 3 9 3 7 9 7 9 7 3 7 2 ) q [ 3 ] ;u3 ( 1 1 . 7 8 0 9 7 2 4 5 0 9 6 2 , 0 , 0) q [ 2 ] ;u3 ( 7 . 8 5 3 9 8 1 6 3 3 9 7 4 , − p i /2 , p i /2) q [ 2 ] ;u3 ( 7 . 8 5 3 9 8 1 6 3 3 9 7 4 , − p i /2 , p i /2) q [ 0 ] ;u3 ( 7 . 8 5 3 9 8 1 6 3 3 9 7 4 , 0 , 0) q [ 1 ] ;u3 ( 8 . 6 3 9 3 7 9 7 9 7 3 7 2 , 0 , 0) q [ 0 ] ;cx q [ 1 ] , q [ 0 ] ;u3 ( 1 1 . 7 8 0 9 7 2 4 5 0 9 6 2 , 0 , 0) q [ 1 ] ;u1 ( 9 . 4 2 4 7 7 7 9 6 0 7 6 9 ) q [ 1 ] ;u1 ( 7 . 0 6 8 5 8 3 4 7 0 5 7 7 ) q [ 0 ] ;u3 ( 9 . 4 2 4 7 7 7 9 6 0 7 6 9 , − p i /2 , p i /2) q [ 0 ] ;cx q [ 1 ] , q [ 0 ] ;u3 ( 1 0 . 9 9 5 5 7 4 2 8 7 5 6 4 , 0 , 0) q [ 1 ] ;u1 ( 1 0 . 2 1 0 1 7 6 1 2 4 1 6 7 ) q [ 1 ] ;u3 ( 1 0 . 2 1 0 1 7 6 1 2 4 1 6 7 , 0 , 0) q [ 0 ] ;u3 ( 7 . 8 5 3 9 8 1 6 3 3 9 7 4 , − p i /2 , p i /2) q [ 0 ] ; ∗ [email protected] † [email protected][1] D. 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