Abstract
Quantum computing promises to offer substantial speedups over its classical counterpart for certain problems. However, the greatest impediment to realizing its full potential is noise that is inherent to these systems. The widely accepted solution to this challenge is the implementation of faulttolerant quantum circuits, which is out of reach for current processors. Here we report experiments on a noisy 127qubit processor and demonstrate the measurement of accurate expectation values for circuit volumes at a scale beyond bruteforce classical computation. We argue that this represents evidence for the utility of quantum computing in a prefaulttolerant era. These experimental results are enabled by advances in the coherence and calibration of a superconducting processor at this scale and the ability to characterize^{1} and controllably manipulate noise across such a large device. We establish the accuracy of the measured expectation values by comparing them with the output of exactly verifiable circuits. In the regime of strong entanglement, the quantum computer provides correct results for which leading classical approximations such as purestatebased 1D (matrix product states, MPS) and 2D (isometric tensor network states, isoTNS) tensor network methods^{2,3} break down. These experiments demonstrate a foundational tool for the realization of nearterm quantum applications^{4,5}.
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Main
It is almost universally accepted that advanced quantum algorithms such as factoring^{6} or phase estimation^{7} will require quantum error correction. However, it is acutely debated whether processors available at present can be made sufficiently reliable to run other, shorterdepth quantum circuits at a scale that could provide an advantage for practical problems. At this point, the conventional expectation is that the implementation of even simple quantum circuits with the potential to exceed classical capabilities will have to wait until more advanced, faulttolerant processors arrive. Despite the tremendous progress of quantum hardware in recent years, simple fidelity bounds^{8} support this bleak forecast; one estimates that a quantum circuit 100 qubits wide by 100 gatelayers deep executed with 0.1% gate error yields a state fidelity less than 5 × 10^{−4}. Nonetheless, the question remains whether properties of the ideal state can be accessed even with such low fidelities. The errormitigation^{9,10} approach to nearterm quantum advantage on noisy devices exactly addresses this question, that is, that one can produce accurate expectation values from several different runs of the noisy quantum circuit using classical postprocessing.
Quantum advantage can be approached in two steps: first, by demonstrating the ability of existing devices to perform accurate computations at a scale that lies beyond bruteforce classical simulation, and second by finding problems with associated quantum circuits that derive an advantage from these devices. Here we focus on taking the first step and do not aim to implement quantum circuits for problems with proven speedups.
We use a superconducting quantum processor with 127 qubits to run quantum circuits with up to 60 layers of twoqubit gates, a total of 2,880 CNOT gates. General quantum circuits of this size lie beyond what is feasible with bruteforce classical methods. We thus first focus on specific test cases of the circuits permitting exact classical verification of the measured expectation values. We then turn to circuit regimes and observables in which classical simulation becomes challenging and compare with results from stateoftheart approximate classical methods.
Our benchmark circuit is the Trotterized time evolution of a 2D transversefield Ising model, sharing the topology of the qubit processor (Fig. 1a). The Ising model appears extensively across several areas in physics and has found creative extensions in recent simulations exploring quantum manybody phenomena, such as time crystals^{11,12}, quantum scars^{13} and Majorana edge modes^{14}. As a test of utility of quantum computation, however, the time evolution of the 2D transversefield Ising model is most relevant in the limit of large entanglement growth in which scalable classical approximations struggle.
In particular, we consider time dynamics of the Hamiltonian,
in which J > 0 is the coupling of nearestneighbour spins with i < j and h is the global transverse field. Spin dynamics from an initial state can be simulated by means of firstorder Trotter decomposition of the timeevolution operator,
in which the evolution time T is discretized into T/δt Trotter steps and \({{\rm{R}}}_{{Z}_{i}{Z}_{j}}({\theta }_{J})\) and \({{\rm{R}}}_{{X}_{i}}({\theta }_{{\rm{h}}})\) are ZZ and X rotation gates, respectively. We are not concerned with the model error owing to Trotterization and thus take the Trotterized circuit as ideal for any classical comparison. For experimental simplicity, we focus on the case θ_{J} = −2Jδt = −π/2 such that the ZZ rotation requires only one CNOT,
where the equality holds up to a global phase. In the resulting circuit (Fig. 1a), each Trotter step amounts to a layer of singlequbit rotations, R_{X}(θ_{h}), followed by commuting layers of parallelized twoqubit rotations, R_{ZZ}(θ_{J}).
For the experimental implementation, we primarily used the IBM Eagle processor ibm_kyiv, composed of 127 fixedfrequency transmon qubits^{15} with heavyhex connectivity and median T_{1} and T_{2} times of 288 μs and 127 μs, respectively. These coherence times are unprecedented for superconducting processors of this scale and allow the circuit depths accessed in this work. The twoqubit CNOT gates between neighbours are realized by calibrating the crossresonance interaction^{16}. As each qubit has at most three neighbours, all ZZ interactions can be performed in three layers of parallelized CNOT gates (Fig. 1b). The CNOT gates within each layer are calibrated for optimal simultaneous operation (see Methods for more details).
We now see that these hardware performance improvements enable even larger problems to be successfully executed with error mitigation, in comparison with recent work^{1,17} on this platform. Probabilistic error cancellation (PEC)^{9} has been shown^{1} to be very effective at providing unbiased estimates of observables. In PEC, a representative noise model is learned and effectively inverted by sampling from a distribution of noisy circuits related to the learned model. Yet, for the current error rates on our device, the sampling overhead for the circuit volumes considered in this work remains restrictive, as discussed further below.
We therefore turn to zeronoise extrapolation (ZNE)^{9,10,17,18}, which provides a biased estimator at a potentially much lower sampling cost. ZNE is either a polynomial^{9,10} or exponential^{19} extrapolation method for noisy expectation values as a function of a noise parameter. This requires the controlled amplification of the intrinsic hardware noise by a known gain factor G to extrapolate to the ideal G = 0 result. ZNE has been widely adopted in part because noiseamplification schemes based on pulse stretching^{9,17,18} or subcircuit repetition^{20,21,22} have circumvented the need for precise noise learning, while relying on simplistic assumptions about the device noise. More precise noise amplification can, however, enable substantial reductions in the bias of the extrapolated estimator, as we demonstrate here.
The sparse Pauli–Lindblad noise model proposed in ref. ^{1} turns out to be especially well suited for noise shaping in ZNE. The model takes the form \({{\rm{e}}}^{{\mathcal{L}}}\), in which \({\mathcal{L}}\) is a Lindbladian comprising Pauli jump operators P_{i} weighted by rates λ_{i}. It was shown in ref. ^{1} that restricting to jump operators acting on local pairs of qubits yields a sparse noise model that can be efficiently learned for many qubits and that accurately captures the noise associated with layers of twoqubit Clifford gates, including crosstalk, when combined with random Pauli twirls^{23,24}. The noisy layer of gates is modelled as a set of ideal gates preceded by some noise channel Λ. Thus, applying Λ^{α} before the noisy layer produces an overall noise channel Λ^{G} with gain G = α + 1. Given the exponential form of the Pauli–Lindblad noise model, the map \({{\rm{e}}}^{\alpha {\mathcal{L}}}\) is obtained by simply multiplying the Pauli rates λ_{i} by α. The resulting Pauli map can be sampled to obtain appropriate circuit instances; for α ≥ 0, the map is a Pauli channel that can be sampled directly, whereas for α < 0, quasiprobabilistic sampling is needed with sampling overhead γ^{−2α} for some modelspecific γ. In PEC, we choose α = −1 to obtain an overall zerogain noise level. In ZNE, we instead amplify the noise^{10,25,26,27} to different gain levels and estimate the zeronoise limit using extrapolation. For practical applications, we need to consider the stability of the learned noise model over time (Supplementary Information III.A), for instance, owing to qubit interactions with fluctuating microscopic defects known as twolevel systems^{28}.
Clifford circuits serve as useful benchmarks of estimates produced by error mitigation, as they can be efficiently simulated classically^{29}. Notably, the entire Ising Trotter circuit becomes Clifford when θ_{h} is chosen to be a multiple of π/2. As a first example, we therefore set the transverse field to zero (R_{X}(0) = I) and evolve the initial state 0⟩^{⊗127} (Fig. 1a). The CNOT gates nominally leave this state unchanged, so the ideal weight1 observables Z_{q} all have expectation value 1; owing to the Pauli twirling of each layer, the bare CNOTs do affect the state. For each Trotter experiment, we first characterized the noise models Λ_{l} for the three Paulitwirled CNOT layers (Fig. 1c) and then used these models to implement Trotter circuits with noise gain levels G ∈ {1, 1.2, 1.6}. Figure 2a illustrates the estimation of ⟨Z_{106}⟩ after four Trotter steps (12 CNOT layers). For each G, we generated 2,000 circuit instances in which, before each layer l, we have inserted products of onequbit and twoqubit Pauli errors i from \({\mathcal{L}}\) drawn with probabilities \({p}_{l,i}=(1{{\rm{e}}}^{2(G1){\lambda }_{l,i}})/2\) and executed each instance 64 times, totalling 384,000 executions. As more circuit instances are accumulated, the estimates of ⟨Z_{106}⟩_{G}, corresponding to the different gains G, converge to distinct values. The different estimates are then fit by an extrapolating function in G to estimate the ideal value ⟨Z_{106}⟩_{0}. The results in Fig. 2a highlight the reduced bias from exponential extrapolation^{19} in comparison with linear extrapolation. That said, exponential extrapolation can exhibit instabilities, for instance, when expectation values are unresolvably close to zero, and—in such cases—we iteratively downgrade the extrapolation model complexity (see Supplementary Information II.B). The procedure outlined in Fig. 2a was applied to the measurement results from each qubit q to estimate all N = 127 Pauli expectations ⟨Z_{q}⟩_{0}. The variation in the unmitigated and mitigated observables in Fig. 2b is indicative of the nonuniformity in the error rates across the entire processor. We report the global magnetization along \(\hat{z}\), \({M}_{z}={\sum }_{q}\,\langle {Z}_{q}\rangle /N\), for increasing depth in Fig. 2c. Although the unmitigated result shows a gradual decay from 1 with an increasing deviation for deeper circuits, ZNE greatly improves agreement, albeit with a small bias, with the ideal value even out to 20 Trotter steps, or 60 CNOT depth. Notably, the number of samples used here is much smaller than an estimate of the sampling overhead that would be needed in a naive PEC implementation (see Supplementary Information IV.B). In principle, this disparity may be greatly reduced by more advanced PEC implementations using lightcone tracing^{30} or by improvements in hardware error rates. As future hardware and software developments bring down sampling costs, PEC may be preferred when affordable to avoid the potentially biased nature of ZNE.
Next, we test the efficacy of our methods for nonClifford circuits and the Clifford θ_{h} = π/2 point, with nontrivial entangling dynamics compared with the identityequivalent circuits discussed in Fig. 2. The nonClifford circuits are of particular importance to test, as the validity of exponential extrapolation is no longer guaranteed (see Supplementary Information V and ref. ^{31}). We restrict the circuit depth to five Trotter steps (15 CNOT layers) and judiciously choose observables that are exactly verifiable. Figure 3 shows the results as θ_{h} is swept between 0 and π/2 for three such observables of increasing weight. Figure 3a shows M_{z} as before, an average of weight1 ⟨Z⟩ observables, whereas Fig. 3b,c show weight10 and weight17 observables. The latter operators are stabilizers of the Clifford circuit at θ_{h} = π/2, obtained by evolution of the initial stabilizers Z_{13} and Z_{58}, respectively, of 0⟩^{⊗127} for five Trotter steps, ensuring nonvanishing expectation values in the strongly entangling regime of particular interest. Although the entire 127qubit circuit is executed experimentally, lightcone and depthreduced (LCDR) circuits enable bruteforce classical simulation of the magnetization and weight10 operator at this depth (see Supplementary Information VII). Over the full extent of the θ_{h} sweep, the errormitigated observables show good agreement with the exact evolution (see Fig. 3a,b). However, for the weight17 operator, the light cone expands to 68 qubits, a scale beyond bruteforce classical simulation, so we turn to tensor network methods.
Tensor networks have been widely used to approximate and compress quantum state vectors that arise in the study of the lowenergy eigenstates of and time evolution by local Hamiltonians^{2,32,33} and, more recently, have been successfully used to simulate lowdepth noisy quantum circuits^{34,35,36}. Simulation accuracy can be improved by increasing the bond dimension χ, which constrains the amount of entanglement of the represented quantum state, at a computational cost scaling polynomially with χ. As entanglement (bond dimension) of a generic state grows linearly (exponentially) with time evolution until it saturates the volume law, deep quantum circuits are inherently difficult for tensor networks^{37}. We consider both quasi1D matrix product states (MPS) and 2D isometric tensor network states (isoTNS)^{3} that have \({\mathcal{O}}({\chi }^{3})\) and \({\mathcal{O}}({\chi }^{7})\) scaling of timeevolution complexity, respectively. Details of both methods and their strengths are provided in Methods and Supplementary Information VI. Specifically for the case of the weight17 operator shown in Fig. 3c, we find that an MPS simulation of the LCDR circuit at χ = 2,048 is sufficient to obtain the exact evolution (see Supplementary Information VIII). The larger causal cone of the weight17 observable results in an experimental signal that is weaker compared with that of the weight10 observable; nevertheless, mitigation still yields good agreement with the exact trace. This comparison suggests that the domain of experimental accuracy could extend beyond the scale of exact classical simulation.
We expect that these experiments will eventually extend to circuit volumes and observables in which such lightcone and depth reductions are no longer important. Therefore, we also study the performance of MPS and isoTNS for the full 127qubit circuit executed in Fig. 3, at respective bond dimensions of χ = 1,024 and χ = 12, which are primarily limited by memory requirements. Figure 3 shows that the tensor network methods struggle with increasing θ_{h}, losing both accuracy and continuity near the verifiable Clifford point θ_{h} = π/2. This breakdown can be understood in terms of entanglement properties of the state. The stabilizer state produced by the circuit at θ_{h} = π/2 has an exactly flat bipartite entanglement spectrum, found from a Schmidt decomposition of a 1D ordering of the qubits. Thus, truncating states with small Schmidt weight—the basis of all tensor network algorithms—is not justified. However, as exact tensor network representations generically require bond dimension exponential in circuit depth, truncation is necessary for tractable numerical simulations.
Finally, in Fig. 4, we stretch our experiments to regimes in which the exact solution is not available with the classical methods considered here. The first example (Fig. 4a) is similar to Fig. 3c but with a further final layer of singlequbit Pauli rotations that interrupt the circuitdepth reduction that previously enabled exact verification for any θ_{h} (see Supplementary Information VII). At the verifiable Clifford point θ_{h} = π/2, the mitigated results agree again with the ideal value, whereas the χ = 3,072 MPS simulation of the 68qubit LCDR circuit markedly fails in the strongly entangling regime of interest. Although χ = 2,048 was sufficient for exact simulation of the weight17 operator in Fig. 3c, an MPS bond dimension of 32,768 would be needed for exact simulation of this modified circuit and operator with θ_{h} = π/2.
As a final example, we extend the circuit depth to 20 Trotter steps (60 CNOT layers) and estimate the θ_{h} dependence of a weight1 observable, ⟨Z_{62}⟩, in Fig. 4b, in which the causal cone extends over the entire device. Given the nonuniformity of device performance, also seen in the spread of singlesite observables in Fig. 2b, we choose an observable that obtains the expected result ⟨Z_{62}⟩ ≈ 1 at the verifiable θ_{h} = 0 point. Despite the greater depth, the MPS simulations of the LCDR circuit agree well with the experiment in the weakly entangling regime of small θ_{h}. Although deviations from the experimental trace emerge with increasing θ_{h}, we note that the MPS simulations slowly move in the direction of the experimental data with increasing χ (see Supplementary Information X) and that the bond dimension needed to exactly represent the stabilizer state and its evolution to depth 20 at θ_{h} = π/2 is 7.2 × 10^{16}, 13 orders of magnitude larger than what we considered (see Supplementary Information VIII). For reference, as the memory required to store an MPS scales as \({\mathcal{O}}({\chi }^{2})\), already a bond dimension of χ = 1 × 10^{8} would require 400 PB, independent of any runtime considerations. Furthermore, fullstate tensor network simulations are already unable to capture the dynamics at the exactly verifiable fivestep circuit in Fig. 3a. We also note that, given the large unmitigated signal, there may be opportunity to study time evolution at even larger depths on the current device.
For execution times, the tensor network simulations in Fig. 4 were run on a 64core, 2.45GHz processor with 128 GB of memory, in which the run time to access an individual data point at fixed θ_{h} was 8 h for Fig. 4a and 30 h for Fig. 4b. The corresponding quantum wallclock run time was approximately 4 h for Fig. 4a and 9.5 h for Fig. 4b, but this is also far from a fundamental limit, being at present dominated by classical processing delays that stand to be largely eliminated through conceptually straightforward optimizations. Indeed, the estimated device run time for the errormitigated expectation values using 614,400 samples (2,400 circuit instances for each gain factor and readout error mitigation, with 64 shots per instance) at a conservative sampling rate of 2 kHz is only 5 min 7 s, which can be even further reduced by optimization of qubit reset speeds. On the other hand, the classical simulations may also be improved by methods besides the purestate tensor networks considered here, such as Heisenberg operator evolution methods, which have recently been applied to nonClifford simulations^{38}. Another approach is to numerically emulate the ZNE used experimentally. For example, it was recently argued that the finiteχ truncation error introduced by tensorproduct compression mimics experimental gate errors^{34}. It would thus be natural to develop a theory for extrapolating tensor network state expectation values in the bond dimension χ for time evolution, as has been done in the case of groundstate search^{39}. Alternatively, one can more directly emulate ZNE by introducing artificial dissipation into the dynamics engineered so that the resulting mixedstate evolution has reduced tensorproduct bond dimension, as—for example—in dissipationassisted operator evolution^{40}, and extrapolate results with respect to the strength of the dissipation. Although such methods^{40,41} can successfully capture the longtime dynamics of the lowweight observables of a 1D spin chain, their applicability to highweight observables in 2D at intermediate times is not clear—particularly as these methods are explicitly constructed to truncate complex operators.
The observation that a noisy quantum processor, even before the advent of faulttolerant quantum computing, produces reliable expectation values at a scale beyond 100 qubits and nontrivial circuit depth leads to the conclusion that there is indeed merit to pursuing research towards deriving a practical computational advantage from noiselimited quantum circuits. Over recent years, substantial research effort has been directed to develop and demonstrate candidate heuristic quantum algorithms^{5} that use noiselimited quantum circuits to estimate expectation values. We have now reached reliability at a scale for which one will be able to verify proposals and explore new approaches to determine which can provide utility beyond classical approximation methods. At the same time, these results will motivate and help advance classical approximation methods as both approaches serve as valuable benchmarks of one another. However, even with improved classical methods, impending orderofmagnitude improvements in gate fidelities^{42} and speed of superconducting quantum systems will drive substantial enhancements in accessible circuit volumes and paint an increasingly bright picture of the utility of noisy quantum computers.
Methods
Device calibration
The speed of crossresonancebased CNOT gates is dependent on the qubit–qubit detuning and, typically, gate speeds across the device are chosen independently to minimize individual gate errors^{43}. This leads to a large spread in CNOT times across the device. Noting that the speed of each parallelized CNOT layer is limited by the slowest gate in the layer, we develop a new tuneup scheme for largescale processor calibration that optimizes the layer rather than the individual gates. First, the control and target qubits are assigned to each gate layer to reduce crosstalk and leakage from transmonfrequency collisions. The slowest gate in each layer then has its duration carefully optimized. Finally, all gates in the layer are fixed to this duration and calibrated simultaneously with erroramplification sequences^{44}. Compared with independently calibrated gates, the layer duration is unchanged, but gates are slower with lower drive amplitudes, reducing any leakage arising from multiphoton transitions. The simultaneous calibration also ensures that the gates are calibrated as they are implemented in the circuit.
Noise model
Throughout this work, we amplify gate noise by means of a learned noise model. For this model, following ref. ^{1}, a general Pauli channel is approximated by \(\Lambda (\rho )=\exp [{\mathcal{L}}](\rho )\) with a sparse Pauli–Lindblad generator
Here the jump operators are chosen to be Pauli operators P_{i} with \({P}_{i}^{\dagger }{P}_{i}=I\) and the model is parameterized by the nonnegative coefficients λ_{i}. This model can be rewritten as
in which \({w}_{i}=(1+{{\rm{e}}}^{2{\lambda }_{i}})/2\) and \({\bigcirc}_{i=1}^{n}{O}_{i}(\,\cdot \,)=({O}_{n}\,\circ \,{O}_{n1}\,\circ \cdots \circ \,{O}_{1})\) represents the composition of operators and O(⋅)(ρ) = O(ρ). In other words, we can express Λ(ρ) as a composition of simple Pauli maps. For physical noise channels, in which all λ_{i} ≥ 0, the composition consists of simply Pauli channels. By allowing nonzero coefficients λ_{i} only for Pauli terms P_{i} whose support corresponds to a single qubit or a pair of connected qubits, we obtain a sparse noise model that can be efficiently learned and that, despite its simplicity, is able to capture crosstalk errors^{1}. It is readily seen that \(\exp [\alpha {\mathcal{L}}]\) is obtained by scaling all λ_{i} by α. For α ≥ 0, the resulting noise model is a composition of Pauli channels. Samples from this channel can be obtained by independently sampling P_{i} with probability 1 − w_{i} for each of the simple channels and multiplying the results. For α < 0, the resulting coefficients 1 − w_{i} are generally negative, leading to a nonphysical noise map. Sampling in that case can still be done, albeit in a quasiprobabilistic manner. Doing so results in a sampling overhead of γ^{2}, in which \(\gamma =\exp \left({\sum }_{i}2{\lambda }_{i}\right)\).
Bruteforce simulations
The simplest, most accurate and most limited method is simulation of a collection of the state of M qubits as a dense vector of 2^{M} complex coefficients. All unitary gates, irrespective of locality, can be applied directly to the vector. Expectation values are found by vector–matrix–vector product of the conjugated state, operator and state. We use this approach for simulations up to 30 qubits.
Tensor network methods
For circuits of more than 30 qubits, we used 1D and 2D tensor network state methods^{45}. For a quantum state on M qubits, tensor network methods approximate the 2^{M} complex coefficients for the wavefunction amplitude as a network of contracted tensors containing \({\mathcal{O}}\left(M{\chi }^{p}\right)\) coefficients, in which p is an integer depending on the method. Here we consider MPS^{2,32,33} with p = 2 and isoTNS^{3} with p = 4. MPS represent a quantum state as a network of rank3 tensors that, when contracted or multiplied together, give an approximation to the wavefunction amplitude for each basis state. isoTNS are a restriction of projected entangled pair states, a 2D generalization of MPS to square lattices in which the network consists of rank5 tensors. The accuracy and computational cost of both MPS and isoTNS depend on the bond dimension χ. MPS methods have the advantage of welldeveloped algorithms, yet suffer from fundamental limitations of using a 1D method to simulate a 2D system. isoTNS methods, on the other hand, are inherently 2D methods but suffer from unavoidable sources of error not present for MPS, though these can be systematically reduced with increasing bond dimension.
Data availability
The datasets generated and analysed during this study are available at https://doi.org/10.6084/m9.figshare.22500355.
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Acknowledgements
This work would not have been possible without the contributions of the entire IBM Quantum team, including: K. Aarhus, B. Abdo, D. W. Abraham, E. Adams, G. Aleksandrowicz, D. Alevras, T. Alexander, A. Alexopoulos, H. Alghassi, L. Ament, M. Amico, A. Anderson, M. Aney, S. A. S. Antezana, J. Apuzzo, E. Arbel, E. Arellano, V. Arena, T. Armon, M. Arthur, P. K. Austel, U. Bacher, S. Bangsaruntip, Z. Barabás, v. barbosa, S. V. Barron, G. Bauer, J. Bauer, M. Beck, M. Beckley, S. W. Bedell, L. Bello, Y. BenHaim, G. Bennett, L. A. Berge, M. Bernagozzi, J. Betke, L. S. Bishop, J. Blair, A. A. Blanco, S. R. Blanks, D. F. Bogorin, R. Bonam, M. Boraas, P. Bosavage, Y. Bosch, S. Bravyi, M. Brink, B. J. Brown, J. S. Broz, J. Bruley, D. Bryant, M. Buehler, M. Byers, M. T. Byrnes, E. Bäumer, C. Cabral Jr., S. Cairns, J. Calderón, L. CarataDejoianu, M. Carlucci, S. Carnevale, A. Carniol, S. Carri, M. S. Carroll, A. Carter, E. C. Castañeda, R. Chadwick, M. E. V. Chan, R. Cheek, V. Chiang, R. 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M.Z. and S.A. were supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Early Career Award No. DESC0022716. Y.W. is supported by the RIKEN iTHEMS fellowship. This work used the Anvil supercomputer at Purdue University through allocation PHY220016 from the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) programme, which is supported by National Science Foundation grant nos. 2138259, 2138286, 2138307, 2137603 and 2138296. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under contract no. DEAC0205CH11231 using NERSC award BESERCAP0024710. This research used the Lawrencium computational cluster resource provided by the IT Division at the Lawrence Berkeley National Laboratory (supported by the Director, Office of Science, Office of Basic Energy Sciences, of the U.S. Department of Energy under contract no. DEAC0205CH11231). We are indebted to Frank Pollmann and ShengHsuan Lin for their contributions to the isoTNS code used in this work. S.A. thanks J. Hauschild for insightful conversations and support with the TeNPy library^{33}, which was used to implement the MPS simulations.
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Y.K. and A.E. ran hardware and numerical experiments. S.A., Y.W. and M.Z. developed the tensor network methods and ran the classical simulations. Y.K., K.X.W. and A.K. developed the devicecalibration protocol and calibrated the quantum processor with S.R. and H.N. E.v.d.B. and K.T. developed the errormitigation protocol. Y.K., A.E., S.A., M.Z., K.T. and A.K. analysed the data. All authors contributed to the manuscript. K.T. and A.K. designed the project. A.K. managed the project.
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Kim, Y., Eddins, A., Anand, S. et al. Evidence for the utility of quantum computing before fault tolerance. Nature 618, 500–505 (2023). https://doi.org/10.1038/s41586023060963
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DOI: https://doi.org/10.1038/s41586023060963
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