Demonstrating quantum error mitigation on logical qubits

A long-standing challenge in quantum computing is developing technologies to overcome the inevitable noise in qubits. To enable meaningful applications in the early stages of fault-tolerant quantum computing, devising methods to suppress post-correction logical failures is becoming increasingly crucial. In this work, we propose and experimentally demonstrate the application of zero-noise extrapolation, a practical quantum error mitigation technique, to error correction circuits on state-of-the-art superconducting processors. By amplifying the noise on physical qubits, the circuits yield outcomes that exhibit a predictable dependence on noise strength, following a polynomial function determined by the code distance. This property enables the effective application of polynomial extrapolation to mitigate logical errors. Our experiments demonstrate a universal reduction in logical errors across various quantum circuits, including fault-tolerant circuits of repetition and surface codes. We observe a favorable performance in multi-round error correction circuits, indicating that this method remains effective when the circuit depth increases. These results advance the frontier of quantum error suppression technologies, opening a practical way to achieve reliable quantum computing in the early fault-tolerant era.

💡 Research Summary

This paper addresses the pressing problem of mitigating noise in quantum processors during the early fault‑tolerant (FT) era, where the number of physical qubits is still limited and full‑scale quantum error correction (QEC) is not yet practical. The authors propose and experimentally demonstrate the integration of zero‑noise extrapolation (ZNE), a leading quantum error mitigation (QEM) technique, with QEC circuits on state‑of‑the‑art superconducting processors.

The central theoretical insight is that, for any quantum circuit—including those that contain mid‑circuit state preparation, measurement, feed‑forward, and post‑selection—the expectation value of an observable ⟨O⟩ as a function of a controllable noise‑scaling factor r can be expressed as a polynomial. In a stochastic error model where each gate fails with probability p, amplifying the error probability to r·p yields
⟨O⟩(r)=⟨O⟩ideal+∑{k=⌈d/2⌉}^{N} a_k r^k,
where d is the effective code distance of the error‑correcting code and N is the total number of noisy operations. Because a QEC code can correct any set of up to ⌈d/2⌉−1 errors, the coefficients a₁,…,a_{⌈d/2⌉−1} vanish, and the leading non‑zero term appears at order r^{⌈d/2⌉}. This contrasts with conventional ZNE on NISQ circuits, where the leading term is linear (k=1). Consequently, the appropriate fitting function for ZNE depends on whether error correction is employed and on the code distance.

Experimentally, the authors use two superconducting quantum processors, each comprising a lattice of tens of frequency‑tunable transmons with adjustable nearest‑neighbor couplings. Median T₁ times exceed 100 µs, CZ gate fidelities are ≈0.995, and readout fidelities are ≈0.995 (Processor I) and 0.991 (Processor II). Noise scaling is achieved by either pulse stretching or by inserting Pauli error channels (I, X, Y, Z) with probabilities proportional to r·p, where p≈3.6 % in the demonstrations.

The first demonstration involves a five‑qubit circuit with a feedback X gate that corrects a bit‑flip on qubit Q₀. By varying r (1, 3) and measuring ⟨Z₀⟩ for different input rotation angles θ₀, the authors show that the expectation value follows the predicted polynomial dependence. With error correction, the decay of ⟨Z₀⟩ with r is markedly slower, confirming the suppression of low‑order error terms. Extrapolation to r=0 using only two data points yields results that match numerical simulations, proving that ZNE works on logical‑level observables.

The core of the paper focuses on fault‑tolerant repetition codes of distances d=3, 5, 7 (up to 13 physical qubits) and on surface‑code patches. For each code, the authors perform M=1–4 rounds of parity‑check measurements, inject Pauli errors at a fixed rate, and evaluate the logical Z operator expectation ⟨Z_L⟩. As r increases, ⟨Z_L⟩ decreases polynomially; however, larger d leads to a flatter curve, indicating operation below the error‑threshold. Applying ZNE with K+1 noise‑scaled points (K=1 or 2) and fitting the appropriate polynomial (starting at order r^{⌈d/2⌉}) yields a mitigated expectation ⟨Z_L⟩_em that is dramatically closer to the ideal value (bias δ = |⟨Z_L⟩_em−1| ≲ 10⁻²).

Two performance metrics are introduced: (i) bias δ, quantifying systematic error after mitigation, and (ii) sampling overhead η = Var