Abstract Unitary errors, such as those arising from fault-tolerant (FT) compilation of quantum algorithms, systematically bias observable estimates. Correcting this bias typically requires additional resources, such as an increased number of non-Clifford gates. In this work, we present an alternative method for correcting bias in the expectation values of observables. The method leverages a decomposition of the ideal quantum channel into a probabilistic mixture of noisy quantum channels. Using this decomposition, we construct unbiased estimators as weighted sums of expectation values obtained from the noisy channels. We provide a detailed analysis of the method, identify the conditions under which it is effective, and validate its performance through numerical simulations. In particular, we demonstrate unbiased observable estimation in the presence of unitary errors by simulating the time dynamics of the Ising Hamiltonian. Our strategy offers a resource-efficient way to reduce the impact of unitary errors, improving methods for estimating observables in noisy near-term quantum devices and FT implementation of quantum algorithms.
Abstract Unitary errors, such as those arising from fault-tolerant (FT) compilation of quantum algorithms, systematically bias observable estimates. Correcting this bias typically requires additional resources, such as an increased number of non-Clifford gates. In this work, we present an alternative method for correcting bias in the expectation values of observables. The method leverages a decomposition of the ideal quantum channel into a probabilistic mixture of noisy quantum channels. Using this decomposition, we construct unbiased estimators as weighted sums of expectation values obtained from the noisy channels. We provide a detailed analysis of the method, identify the conditions under which it is effective, and validate its performance through numerical simulations. In particular, we demonstrate unbiased observable estimation in the presence of unitary errors by simulating the time dynamics of the Ising Hamiltonian. Our strategy offers a resource-efficient way to reduce the impact of unitary errors, improving methods for estimating observables in noisy near-term quantum devices and FT implementation of quantum algorithms.
Modeling and simulation are essential for predicting and verifying the behavior of fabricated quantum circuits, but existing simulation methods are either impractically costly or require an unrealistic simplification of error processes. We present a method of simulating noisy Clifford circuits that is both accurate and practical in experimentally relevant regimes. In particular, the cost is weakly exponential in the size and the degree of non-Cliffordness of the circuit. Our approach is based on the construction of exact representations of quantum channels as quasiprobability distributions over stabilizer operations, which are then sampled, simulated, and weighted to yield unbiased statistical estimates of circuit outputs and other observables. As a demonstration of these techniques, we simulate a Steane [[7,1,3]]-encoded logical operation with non-Clifford errors and compute its fault tolerance error threshold. We expect that the method presented here will enable studies of much larger and more realistic quantum circuits than was previously possible.
Fault-tolerant quantum computation (FTQC) is essential to implement quantum algorithms in a noise-resilient way, and thus to enjoy advantages of quantum computers even with presence of noise. In FTQC, a quantum circuit is decomposed into universal gates that can be fault-tolerantly implemented, for example, Clifford+ T gates. Here, T gate is usually regarded as an essential resource for quantum computation because its action cannot be simulated efficiently on classical computers and it is experimentally difficult to implement fault-tolerantly. Practically, it is highly likely that only a limited number of T gates are available in the near future. Pre-FTQC era, due to the constraint on available resources, it is vital to precisely estimate the decomposition error of a whole circuit. In this paper, we propose that the Clifford+ T decomposition error for a given quantum circuit containing a large number of quantum gates can be modeled as the depolarizing noise by averaging the decomposition error for each quantum gate in the circuit, and our model provides more accurate error estimation than the naive estimation. We exemplify this by taking unitary coupled-cluster (UCC) ansatz used in the applications of quantum computers to quantum chemistry as an example. We theoretically evaluate the approximation error of UCC ansatz when decomposed into Clifford+ T gates, and the numerical simulation for a wide variety of molecules verified that our model well explains the total decomposition error of the ansatz. Our results enable the precise and efficient usage of quantum resources in the early-stage applications of quantum computers and fuel further research towards what quantum computation can achieve in the upcoming future.
Accurately estimating observables on noisy quantum devices remains a central challenge for near-term quantum algorithms. While quantum error mitigation techniques can reduce noise-induced bias, they often rely on unverifiable assumptions about the circuit noise, and cannot guarantee the magnitude of residual bias error. Here, rather than using classical resources to mitigate a noisy quantum circuit execution, we propose a hybrid algorithm that uses quantum resources to improve the accuracy of approximate classical Pauli-path simulation. Our protocol, Quantum Enhanced Pauli Propagation (QuEPP), uses Clifford perturbation theory (CPT) to construct a classically simulable ensemble of Clifford circuits from the low-order terms in CPT, which directly provide the approximate classical Pauli-path simulation of the target circuit. Noisy quantum expectation values of this ensemble are then used to infer a global rescaling factor that corrects quantum execution of the target circuit, providing higher-order contributions absent from the truncated low-order classical simulation. This approach requires no noise characterization, applies to arbitrary circuits, and provides a provable route to asymptotically unbiased estimates. Using IBM Heron hardware, we demonstrate QuEPP on 2D random mirror circuits of up to 49 qubits and circuit depth 80, as well as Trotterized Hamiltonian evolution, showing consistent improvements beyond classical CPT and unmitigated quantum results. QuEPP offers a simple, scalable, and model-free framework for enabling accurate quantum computation in the pre-fault-tolerant era.
When estimating the eigenvalues of a given observable, even fault-tolerant quantum computers will be subject to errors, namely algorithmic errors. These stem from approximations in the algorithms implementing the unitary passed to phase estimation to extract the eigenvalues, e.g. Trotterisation or qubitisation. These errors can be tamed by increasing the circuit complexity, which may be unfeasible in early-stage fault-tolerant devices. Rather, we propose in this work an error mitigation strategy that enables a reduction of the algorithmic errors up to any order, at the cost of evaluating the eigenvalues of a set of observables implementable with limited resources. The number of required observables is estimated and is shown to only grow polynomially with the number of terms in the Hamiltonian, and in some cases, linearly with the desired order of error mitigation. Our results show error reduction of several orders of magnitude in physically relevant cases, thus promise accurate eigenvalue estimation even in early fault-tolerant devices with limited number of qubits.
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 size and code distance.
Fault-tolerant quantum computation enables the deployment of practical quantum algorithms but incurs substantial overhead from error correction, making resource estimation a central concern. Beyond case-by-case analyses, existing quantum programming languages either require programmers to manipulate low-level hardware details, rendering fault-tolerant implementations cumbersome, or abstract away the underlying error-correction schemes, reducing the effectiveness of resource utilization and estimation.
To address these limitations while preserving programmability, we present a quantum programming language that enables efficient resource utilization, together with a resource-estimation framework for comprehensive resource analysis. Our framework features programmer-visible abstractions of error-correction schemes and cross-layer program-hardware analysis, allowing systematic exploration of resource trade-offs. We evaluate our approach on detailed fault-tolerant implementations of practical large-scale quantum algorithms, including components typically treated as black boxes in existing frameworks. The results demonstrate that our framework enables substantial resource savings while delivering detailed, fine-grained, and accurate resource estimates for fault-tolerant quantum programs.