Compilation-informed probabilistic logical-error cancellation

TL;DR

This work introduces a QEM scheme against both compilation errors and logical-gate noise that is circuit-, QEC code-, and compiler-agnostic and significantly reduces quantum resource requirements for high-precision estimations, offering a practical route towards fault-tolerant quantum computation with precision-independent overheads for fixed circuit complexity and code distance.

Abstract

The potential of quantum computers to outperform classical ones in practically useful tasks remains challenging in the near term due to scaling limitations and high error rates of current quantum hardware. While quantum error correction (QEC) offers a clear path towards fault tolerance, overcoming the scalability issues will take time. Early applications will likely rely on QEC combined with quantum error mitigation (QEM). We introduce a QEM scheme against both compilation errors and logical-gate noise that is circuit-, QEC code-, and compiler-agnostic. The scheme builds on quasi-probability methods and uses information about the circuit's gates' compilations to attain an unbiased estimation of noiseless expectation values incurring a constant sample-complexity overhead. Moreover, it features maximal circuit size and code distance both independent of the target precision, in contrast to strategies based on QEC alone. We formulate the mitigation procedure as a linear program, demonstrate its efficacy through numerical simulations, and illustrate it for estimating the Jones polynomials of knots. Our method significantly reduces quantum resource requirements for high-precision estimations, offering a practical route towards fault-tolerant quantum computation with precision-independent overheads for fixed circuit complexity and code distance.

Figures (5)

• Figure 1: Compilation-informed probabilistic error cancellation. Upper panel: Given a characterized noisy logical quantum device, one builds a basis for the space of 2-qubit unitary channels, which includes sequences (pink boxes) of noisy logical operations (red boxes) from a universal set. Lower panel: To find an $\varepsilon$-precise estimate of the expectation value $\langle O\rangle$ of an observable $O$ on an input state $\rho$ evolved by a quantum circuit $U$, each 2-qubit gate $U_i$ (blue boxes) in $U$ is first compiled up to $\varepsilon$-independent precision into a sequence (golden lozenges) of ideal unitary gates from the universal set and then decomposed as an affine combination of basis channels and the noisy version of that compilation sequence. The decomposition coefficients are treated as quasi-probabilities, with the noisy compiled sequence being the dominant term. $\langle O\rangle$ is then estimated statistically by Monte-Carlo sampling noisy circuits from the distribution defined by them. As a result, each circuit's depth and the required QEC code-distance are both independent of $\varepsilon$.
• Figure 2: Maximum feasible circuit size. Solid lines depict (in loglog scale) the maximum feasible circuit size for CIPEC after compilation, $L_\text{max}=1/(2c_*\epsilon_\mathtt{Q})$, as a function of the logical error rate. The sample overhead is $\gamma^2=e^2$ and the bases are $\widetilde{B}_1,\widetilde{B}_2,\widetilde{B}_3$ of Tab. \ref{['tab:bases']}. These curves remain the same regardless of the precision $1/\varepsilon$. Dashed lines show the corresponding quantity for a strategy without QEM where compilation, systematic, and statistical errors are taken equal (i.e., choosing $\xi=\eta=3$ in Lemma \ref{['lemma:maxG_ECstrategy']} in App. \ref{['app:proofs']}), in which case $L_\text{max}=\varepsilon/(3\epsilon_\mathtt{Q})$. One sees that, for any fixed $\epsilon_\texttt{Q}$, CIPEC allows solving instances not solvable by QEC alone -- namely, those requiring precision $1/\varepsilon \geq 2c_*/3$. One explicit such example is shown in Fig. \ref{['fig:jones']}.
• Figure 3: CIPEC for Jones polynomial estimation. Quantum estimation of the Jones polynomial at $q=e^{i \frac{2\pi}{5}}$ for the trefoil knot (inset) using the control-free Hadamard test of laakkonen_less_2025. The target relative precision is $\varepsilon=10^{-2}$ and we use QiboQibo2021 to simulate a device of $n_{\ell}=5$ noisy logical qubits. For convenience, we display the results in terms of the scaled quantity $\bra{s}U_{\Sigma}\ket{s}\propto J_{\text{trefoil}}(e^{i \frac{2\pi}{5}})$ defined in Eq.(E1) of SI-V. The blue star marks the exact result $J_{\text{trefoil}}(q)=q^{-1}+q^{-3}-q^{-5}$; green triangles are the result of 10 distinct trial estimations without error mitigation using a shot-noise simulation of the compiled circuit for $U_{\Sigma}$; the orange dots show the result of 10 distinct trials using CIPEC with the minimal basis $\widetilde{B}_2$ and a sample overhead $\gamma^2\approx2.46$; the number of samples in both cases follows the expressions in SI-II with a failure probability $\delta=0.1$. We see that CIPEC consistently delivers estimates within the required precision while the unmitigated estimates fail.
• Figure 4: (Two-qubit gate decomposition used in our Clifford+T compilation) Decomposition of an arbitrary two-qubit unitary in terms of $15\,R_z(\theta$), $10\,\sqrt{X}$, and $3\,\mathrm{CNOT}$ gates. Each $\sqrt{X}$ is exactly compiled into Clifford+T as $\sqrt{\mathrm{X}}=e^{i\pi/4}\mathrm{S}^\dagger\mathrm{H}\mathrm{S}^\dagger$, while $R_z$ gates are approximately compiled to any target precision using gridsynthross2016.
• Figure 5: (Clifford+T compilation scaling). The points show the total number of gates $L$ (in $\log$-scale) vs. the inverse compilation error $\epsilon_c$ (in $\log\log$-scale) for $10^4$ Haar random 2-qubit unitaries $U_\text{Haar}$. Each $U_\text{Haar}$ is first decomposed as in Fig. \ref{['fig:2q-rz-sx']} and then synthesized into Clifford+T gates using gridsynth with a target compilation error uniformly sampled (in log-log scale) from $[10^{-9},10^{-2}]$. Assuming the Solovay-Kitaev-like polylogarithmic scaling \ref{['eq:skscaling']} in the average case, we determine the constants $c_1,c_2$ through a linear fit (orange curve). For comparison, we show how these values change when the fit is restricted to the asymptotic regime (green curve) or to the low precision regime (red curve).

Theorems & Definitions (19)

• Definition 2: Noisy logical quantum device
• Definition 3: CIPEC-capable device
• Definition 4: Worst-case negativity
• Theorem 6: CIPEC
• Lemma 7: Negativity and the diamond norm
• proof
• Lemma 8: Two-qubit Unitary Synthesis using noisy channels
• proof
• Lemma 9: CIPEC negativity upper bound
• proof