Evidence for the utility of quantum computing before fault tolerance

The study addresses whether today’s noisy quantum processors, without fault tolerance, can still deliver accurate expectation values for non-trivial circuits at scales beyond brute-force classical simulation. Although many advanced quantum algorithms will require error correction, it is debated if near-term devices can reliably execute shorter-depth circuits to offer practical advantage. The authors propose and test an error-mitigation approach to extract accurate observables from noisy runs of large circuits, focusing on demonstrating reliable computation at scale rather than implementing algorithms with proven speedups. They target Trotterized time evolution of a 2D transverse-field Ising model on a 127-qubit superconducting processor and ask if accurate, verifiable observables can be obtained even when state fidelity is low, particularly in strongly entangling regimes where classical tensor-network approximations struggle.

The work builds on error mitigation techniques for noisy intermediate-scale quantum (NISQ) devices, particularly probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE). Prior studies established PEC as an unbiased but sampling-costly method and ZNE as a biased but practical approach via noise amplification (e.g., pulse stretching or circuit repetition). A sparse Pauli-Lindblad noise model was recently shown to efficiently capture noise in layers of two-qubit Clifford gates, enabling precise noise shaping. Classical simulation baselines include efficient stabilizer/Clifford simulation and tensor networks (MPS, isoTNS) widely used for time evolution in local Hamiltonians, with limitations in highly entangling regimes due to bond-dimension growth. The authors position their contribution relative to earlier demonstrations of error mitigation on smaller systems and discuss complementary classical advances (e.g., low-rank stabilizer decompositions, operator-evolution methods) and light-cone/depth-reduction techniques for exact verification where possible.

Hardware and circuits: Experiments were performed on the 127-qubit IBM Eagle processor (ibm_kyiv) with heavy-hex connectivity and median coherence times T1 = 288 μs and T2 = 127 μs. Two-qubit CNOT gates are realized via the cross-resonance interaction and arranged in three parallel layers to cover all nearest neighbors per Trotter step. The benchmark circuit is the first-order Trotterized time evolution of the 2D transverse-field Ising model H = Σ⟨i,j⟩ ZiZj + Σi Xi. Each Trotter step comprises single-qubit Rx(θ) rotations followed by commuting two-qubit RZZ(θ) rotations. For experimental simplicity, θ = −π/2 (for ZZ) is chosen so that RZZ(π/2) can be implemented with a single CNOT (up to global phase). Circuit depths include up to 20 Trotter steps (60 CNOT layers), with as many as 2,880 CNOT gates. Noise characterization and shaping: Random Pauli twirling is applied to each CNOT layer to render the noise approximately Pauli and layer-local. The sparse Pauli-Lindblad noise model e^{Lt} with local (one- and two-qubit) Pauli jump operators Pi and rates λi is learned per CNOT layer, capturing noise and crosstalk efficiently across the device. For each layer l, a Pauli channel Al parameterized by learned rates is used. Noise amplification is done probabilistically: inserting Pauli errors before each noisy layer according to probabilities Pi = (1 − e^{−2(G−1)λi})/2 produces an effective gain G in the overall noise channel; G = 1 is unmitigated, G > 1 amplifies noise for ZNE, and G = 0 is approximated via quasi-probabilistic sampling with a = −1 for PEC (with sampling overhead γ). Exponential and linear extrapolations in G are used to infer the zero-noise limit, with model downgrading if instability is detected (e.g., when observables are near zero). Experimental conditions and sampling: For ZNE at the Clifford condition (θx = 0), gain levels G ∈ {1, 1.2, 1.6} were used. For each G, 2,000 circuit instances were generated per experiment, each executed with 64 shots (e.g., totaling 384,000 executions in Fig. 2a). Across different figures: for Fig. 3 (five Trotter steps), gains G = {1, 1.2, 1.6}; 1,800–2,000 instances per G for magnetization and weight-10 observables, and 2,500–3,000 for the weight-17 observable. For Fig. 4a, G = {1, 1.3, 1.6} with 2,000–3,200 instances per G; for Fig. 4b, G = {1, 1.2, 1.6} with 1,700–2,400 instances per G. Percentile bootstrap provides 68% confidence intervals. Readout-error mitigation is applied. Noise model stability over time was monitored. Verification regimes and observables: Clifford circuits (θx multiples of π/2) serve as exact-verification points via classical stabilizer simulation. Non-Clifford circuits at depth five (15 CNOT layers) are verified where possible using light-cone and depth-reduced (LCDR) circuits enabling brute-force classical evaluation of specific observables (e.g., magnetization Mz; weight-10 operator). A weight-17 observable has a 68-qubit light cone, beyond brute-force, so approximate tensor networks (MPS, isoTNS) are used. Additional tests include modifying circuits (e.g., final single-qubit rotations) to break LCDR exactness and increasing depth to 20 Trotter steps with weight-1 observables whose causal cones span the full device. Classical simulations: Pure-state tensor networks were employed: MPS with bond dimensions χ up to 3,072 for LCDR circuits and up to 1,024 for full 127-qubit circuits; isoTNS with χ = 12 for full circuits. Performance and limitations are benchmarked against experimental (unmitigated and ZNE-mitigated) data. Reported CPU environment: 64-core 2.45 GHz, 128 GB memory; per-point wall-clock ~8 h (Fig. 4a) and ~30 h (Fig. 4b). Quantum run times per point were ~4 h and ~9.5 h but could be reduced to minutes with higher sampling rates and optimized resets; e.g., 614,400 total samples at 2 kHz would take ~5 min.

• Accurate expectation values at 127 qubits and substantial depth: Using ZNE with probabilistic noise amplification informed by a learned sparse Pauli-Lindblad noise model, the processor produced accurate expectation values for circuits with up to 20 Trotter steps (60 CNOT layers, as many as 2,880 CNOTs), at scales beyond brute-force classical simulation.
• Clifford benchmark (θx = 0): For four Trotter steps, ZNE extrapolations (especially exponential) converged to near-ideal values of ⟨Zi⟩ despite unmitigated decay. Magnetization Mz remained close to the ideal value 1 after ZNE even up to 20 steps; unmitigated estimates decayed monotonically with depth. Example sampling: 2,000 instances per gain, 64 shots each (384,000 total executions) were sufficient to resolve distinct gain-dependent expectations and perform stable extrapolations.
• Non-Clifford and verifiable regimes at five steps: For magnetization (weight-1) and a weight-10 observable, ZNE-mitigated results closely tracked exact classical evolution across θ sweeps, while unmitigated traces deviated. For a weight-17 observable with a 68-qubit causal cone, MPS with χ = 2,048 (LCDR) could match exact evolution in Fig. 3c, but tensor networks (MPS χ = 1,024; isoTNS χ = 12) on the full 127-qubit circuit lost accuracy and continuity near θ = π/2.
• Beyond exact verification: Modifying the circuit by adding final single-qubit rotations (Fig. 4a) broke LCDR exactness; ZNE-mitigated results remained accurate at the verifiable point θ = π/2 (expectation −1, plotted negated), while MPS (LCDR, χ = 3,072) failed to approach the ideal. Estimated exact MPS bond dimension required for this case at θ = π/2 is ~32,768. For depth 20 and ⟨Z2⟩ (Fig. 4b), MPS (LCDR, χ = 1,024) agreed at small θ but deviated as entanglement increased, whereas experimental ZNE traces stayed consistent; exact bond dimension needed at θ = π/2 is ~7.2 × 10^6 (memory ~400 PB for χ ≈ 10^6), far beyond practical classical limits.
• Efficiency and scaling: Compared to naive PEC sampling overheads, ZNE required far fewer samples to achieve useful accuracy. Experimental wall-clock per data point was hours due to classical overheads but could be reduced to minutes with higher sampling throughput; classical tensor-network simulations already required many hours per point at modest χ.
• Breakdown of tensor networks near the Clifford point θ = π/2: The circuit produces stabilizer states with flat entanglement spectra, invalidating truncation by small Schmidt weights and causing failure of MPS/isoTNS at feasible bond dimensions. This highlights regimes where quantum experiments yield reliable results while leading classical approximations break down.

The results directly address whether noisy, pre-fault-tolerant quantum processors can deliver reliable expectation values at classically intractable scales. By integrating precise noise characterization with probabilistic noise amplification and ZNE, the experiments demonstrate accurate observables on a 127-qubit device at depths where brute-force classical computation is infeasible and state-of-the-art tensor networks become unreliable in strongly entangling regimes. The agreement with exact solutions in verifiable regimes (Clifford points and LCDR-reduced observables) validates the mitigation pipeline and supports extrapolated results where exact verification is impossible. The findings indicate that, even with low overall state fidelities, carefully designed error mitigation allows access to properties of the ideal state. This establishes a credible path to near-term utility: practical quantum computations focusing on expectation values can outpace classical approximations for certain problem instances before full fault tolerance. The observed classical breakdown near θ = π/2 underscores the value of experiments as benchmarks that can also inform advances in classical methods.

This work demonstrates evidence for the utility of quantum computing before fault tolerance by measuring accurate expectation values on a 127-qubit superconducting processor at substantial circuit depths using error mitigation. Key contributions include: (1) scalable learning of a sparse Pauli-Lindblad noise model across a large device; (2) precise probabilistic noise amplification enabling effective ZNE with reduced bias; (3) validation against exact classical results in verifiable regimes; and (4) successful operation in regimes where leading tensor-network approximations fail. The results provide a foundational toolset for near-term quantum applications that estimate observables, suggesting that practical advantage may be attainable without error correction for selected tasks. Future directions include reducing sampling and runtime overheads (e.g., via faster qubit resets), improving hardware fidelities and gate speeds, exploring advanced PEC with light-cone tracing, expanding classical baselines (operator-evolution methods, ZNE emulation in classical simulations), and extending to deeper circuits and broader classes of observables and problems with industrial relevance.

• ZNE provides a biased estimator and can exhibit instabilities (e.g., when expectations are near zero), requiring model downgrading; PEC is unbiased but may be prohibitively expensive at current error rates.
• The approach depends on learned noise models and their temporal stability; device non-uniformity and drift (e.g., due to two-level systems) can impact accuracy.
• Validation relies on specific observables and verification regimes (Clifford points, LCDR reductions); full-circuit, all-observable exact verification is not feasible at these scales.
• Trotterization error is not addressed (the Trotterized circuit is taken as the ideal reference), so conclusions pertain to reproducing the ideal Trotter model, not the exact continuous-time dynamics.
• Classical baselines are limited to pure-state tensor networks with feasible bond dimensions; alternative classical methods could shift the boundary but are also challenged in strongly entangling 2D regimes.
• Experimental run times included classical processing overheads; while reducible, they currently affect throughput.