```html Authors: Youngseok Kim Andrew Eddins Sajant Anand Ken Xuan Wei Ewout van den Berg Sami Rosenblatt Hasan Nayfeh Yantao Wu Michael Zaletel Kristan Temme Abhinav Kandala Abstract Quantum computing promises to offer substantial speed-ups 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 fault-tolerant quantum circuits, which is out of reach for current processors. Here we report experiments on a noisy 127-qubit processor and demonstrate the measurement of accurate expectation values for circuit volumes at a scale beyond brute-force classical computation. We argue that this represents evidence for the utility of quantum computing in a pre-fault-tolerant 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 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 pure-state-based 1D (matrix product states, MPS) and 2D (isometric tensor network states, isoTNS) tensor network methods , break down. These experiments demonstrate a foundational tool for the realization of near-term quantum applications , . 1 2 3 4 5 Main It is almost universally accepted that advanced quantum algorithms such as factoring or phase estimation will require quantum error correction. However, it is acutely debated whether processors available at present can be made sufficiently reliable to run other, shorter-depth 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, fault-tolerant processors arrive. Despite the tremendous progress of quantum hardware in recent years, simple fidelity bounds support this bleak forecast; one estimates that a quantum circuit 100 qubits wide by 100 gate-layers 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 error-mitigation , approach to near-term 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 post-processing. 6 7 8 9 10 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 brute-force 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 speed-ups. We use a superconducting quantum processor with 127 qubits to run quantum circuits with up to 60 layers of two-qubit gates, a total of 2,880 CNOT gates. General quantum circuits of this size lie beyond what is feasible with brute-force 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 state-of-the-art approximate classical methods. Our benchmark circuit is the Trotterized time evolution of a 2D transverse-field Ising model, sharing the topology of the qubit processor (Fig. ). The Ising model appears extensively across several areas in physics and has found creative extensions in recent simulations exploring quantum many-body phenomena, such as time crystals , , quantum scars and Majorana edge modes . As a test of utility of quantum computation, however, the time evolution of the 2D transverse-field Ising model is most relevant in the limit of large entanglement growth in which scalable classical approximations struggle. 1a 11 12 13 14 , Each Trotter step of the Ising simulation includes single-qubit and two-qubit rotations. Random Pauli gates are inserted to twirl (spirals) and controllably scale the noise of each CNOT layer. The dagger indicates conjugation by the ideal layer. , Three depth-1 layers of CNOT gates suffice to realize interactions between all neighbour pairs on ibm_kyiv. , Characterization experiments efficiently learn the local Pauli error rates , (colour scales) comprising the overall Pauli channel Λ associated with the th twirled CNOT layer. (Figure expanded in Supplementary Information ). , Pauli errors inserted at proportional rates can be used to either cancel (PEC) or amplify (ZNE) the intrinsic noise. a X ZZ b c λl i l l IV.A d In particular, we consider time dynamics of the Hamiltonian, in which > 0 is the coupling of nearest-neighbour spins with < and is the global transverse field. Spin dynamics from an initial state can be simulated by means of first-order Trotter decomposition of the time-evolution operator, J i j h in which the evolution time is discretized into / Trotter steps and and are and 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 = −2 = −π/2 such that the rotation requires only one CNOT, T T δt ZZ X θJ Jδt ZZ where the equality holds up to a global phase. In the resulting circuit (Fig. ), each Trotter step amounts to a layer of single-qubit rotations, R ( h), followed by commuting layers of parallelized two-qubit rotations, R ( ). 1a X θ ZZ θJ For the experimental implementation, we primarily used the IBM Eagle processor ibm_kyiv, composed of 127 fixed-frequency transmon qubits with heavy-hex connectivity and median 1 and 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 two-qubit CNOT gates between neighbours are realized by calibrating the cross-resonance interaction . As each qubit has at most three neighbours, all interactions can be performed in three layers of parallelized CNOT gates (Fig. ). The CNOT gates within each layer are calibrated for optimal simultaneous operation (see for more details). 15 T T 16 ZZ 1b Methods We now see that these hardware performance improvements enable even larger problems to be successfully executed with error mitigation, in comparison with recent work , on this platform. Probabilistic error cancellation (PEC) has been shown 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. 1 17 9 1 We therefore turn to zero-noise extrapolation (ZNE) , , , , which provides a biased estimator at a potentially much lower sampling cost. ZNE is either a polynomial , or exponential 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 to extrapolate to the ideal = 0 result. ZNE has been widely adopted in part because noise-amplification schemes based on pulse stretching , , or subcircuit repetition , , 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. 9 10 17 18 9 10 19 G G 9 17 18 20 21 22 The sparse Pauli–Lindblad noise model proposed in ref. turns out to be especially well suited for noise shaping in ZNE. The model takes the form , in which is a Lindbladian comprising Pauli jump operators weighted by rates . It was shown in ref. 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 two-qubit Clifford gates, including crosstalk, when combined with random Pauli twirls , . The noisy layer of gates is modelled as a set of ideal gates preceded by some noise channel Λ. Thus, applying Λ before the noisy layer produces an overall noise channel Λ with gain = + 1. Given the exponential form of the Pauli–Lindblad noise model, the map is obtained by simply multiplying the Pauli rates by . The resulting Pauli map can be sampled to obtain appropriate circuit instances; for ≥ 0, the map is a Pauli channel that can be sampled directly, whereas for < 0, quasi-probabilistic sampling is needed with sampling overhead −2 for some model-specific . In PEC, we choose = −1 to obtain an overall zero-gain noise level. In ZNE, we instead amplify the noise , , , to different gain levels and estimate the zero-noise limit using extrapolation. For practical applications, we need to consider the stability of the learned noise model over time (Supplementary Information ), for instance, owing to qubit interactions with fluctuating microscopic defects known as two-level systems . 1 Pi λi 1 23 24 α G G α λi α α α γ α γ α 10 25 26 27 III.A 28 Clifford circuits serve as useful benchmarks of estimates produced by error mitigation, as they can be efficiently simulated classically . Notably, the entire Ising Trotter circuit becomes Clifford when h is chosen to be a multiple of π/2. As a first example, we therefore set the transverse field to zero (R (0) = ) and evolve the initial state |0⟩⊗127 (Fig. ). The CNOT gates nominally leave this state unchanged, so the ideal weight-1 observables 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 for the three Pauli-twirled CNOT layers (Fig. ) and then used these models to implement Trotter circuits with noise gain levels ∈ {1, 1.2, 1.6}. Figure illustrates the estimation of ⟨ 106⟩ after four Trotter steps (12 CNOT layers). For each , we generated 2,000 circuit instances in which, before each layer , we have inserted products of one-qubit and two-qubit Pauli errors from drawn with probabilities and executed each instance 64 times, totalling 384,000 executions. As more circuit instances are accumulated, the estimates of ⟨ 106⟩ , corresponding to the different gains , converge to distinct values. The different estimates are then fit by an extrapolating function in to estimate the ideal value ⟨ 106⟩0. The results in Fig. highlight the reduced bias from exponential extrapolation 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 ). The procedure outlined in Fig. was applied to the measurement results from each qubit to estimate all = 127 Pauli expectations ⟨ ⟩0. The variation in the unmitigated and mitigated observables in Fig. is indicative of the non-uniformity in the error rates across the entire processor. We report the global magnetization along , , for increasing depth in Fig. . 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 ). In principle, this disparity may be greatly reduced by more advanced PEC implementations using light-cone tracing 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. 29 θ X I 1a Zq l 1c G 2a Z G l i Z G G G Z 2a 19 II.B 2a q N Zq 2b 2c IV.B 30 Mitigated expectation values from Trotter circuits at the Clifford condition h = 0. , Convergence of unmitigated ( = 1), noise-amplified ( >1) and noise-mitigated (ZNE) estimates of ⟨ 106⟩ after four Trotter steps. In all panels, error bars indicate 68% confidence intervals obtained by means of percentile bootstrap. Exponential extrapolation (exp, dark blue) tends to outperform linear extrapolation (linear, light blue) when differences between the converged estimates of ⟨ 106⟩ ≠0 are well resolved. , Magnetization (large markers) is computed as the mean of the individual estimates of ⟨ ⟩ for all qubits (small markers). , As circuit depth is increased, unmitigated estimates of decay monotonically from the ideal value of 1. ZNE greatly improves the estimates even after 20 Trotter steps (see Supplementary Information for ZNE details). θ a G G Z Z G b Zq c Mz II Next, we test the efficacy of our methods for non-Clifford circuits and the Clifford h = π/2 point, with non-trivial entangling dynamics compared with the identity-equivalent circuits discussed in Fig. . The non-Clifford circuits are of particular importance to test, as the validity of exponential extrapolation is no longer guaranteed (see Supplementary Information and ref. ). We restrict the circuit depth to five Trotter steps (15 CNOT layers) and judiciously choose observables that are exactly verifiable. Figure shows the results as h is swept between 0 and π/2 for three such observables of increasing weight. Figure shows as before, an average of weight-1 ⟨ ⟩ observables, whereas Fig. show weight-10 and weight-17 observables. The latter operators are stabilizers of the Clifford circuit at h = π/2, obtained by evolution of the initial stabilizers 13 and 58, respectively, of |0⟩⊗127 for five Trotter steps, ensuring non-vanishing expectation values in the strongly entangling regime of particular interest. Although the entire 127-qubit circuit is executed experimentally, light-cone and depth-reduced (LCDR) circuits enable brute-force classical simulation of the magnetization and weight-10 operator at this depth (see Supplementary Information ). Over the full extent of the h sweep, the error-mitigated observables show good agreement with the exact evolution (see Fig. ). However, for the weight-17 operator, the light cone expands to 68 qubits, a scale beyond brute-force classical simulation, so we turn to tensor network methods. θ 2 V 31 3 θ 3a Mz Z 3b,c θ Z Z VII θ 3a,b Expectation value estimates for h sweeps at a fixed depth of five Trotter steps for the circuit in Fig. . The considered circuits are non-Clifford except at h = 0, π/2. Light-cone and depth reductions of respective circuits enable exact classical simulation of the observables for all h. For all three plotted quantities (panel titles), mitigated experimental results (blue) closely track the exact behaviour (grey). In all panels, error bars indicate 68% confidence intervals obtained by means of percentile bootstrap. The weight-10 and weight-17 observables in and are stabilizers of the circuit at h = π/2 with respective eigenvalues +1 and −1; all values in have been negated for visual simplicity. The lower inset in depicts variation of ⟨ ⟩ at h = 0.2 across the device before and after mitigation and compares with exact results. Upper insets in all panels illustrate causal light cones, indicating in blue the final qubits measured (top) and the nominal set of initial qubits that can influence the state of the final qubits (bottom). also depends on 126 other cones besides the example shown. Although in all panels exact results are obtained from simulations of only causal qubits, we include tensor network simulations of all 127 qubits (MPS, isoTNS) to help gauge the domain of validity for those techniques, as discussed in the main text. isoTNS results for the weight-17 operator in are not accessible with current methods (see Supplementary Information ). All experiments were carried out for = 1, 1.2, 1.6 and extrapolated as in Supplementary Information . For each , we generated 1,800–2,000 random circuit instances for and and 2,500–3,000 instances for . θ 1a θ θ b c θ c a Zq θ Mz c VI G II.B G a b c Tensor networks have been widely used to approximate and compress quantum state vectors that arise in the study of the low-energy eigenstates of and time evolution by local Hamiltonians
