We propose several optimizations of the CliNR partial error correction scheme which implements Clifford circuits by consuming a resource state. Errors are corrected by measuring a sequence of Pauli operators that we refer to as the verification sequence. We first propose a global optimization algorithm searching for a verification sequence resulting in a low logical error rate using tabu search. Then, we introduce a proxy for the logical error rate which is easier to evaluate and we design a two-step optimization algorithm. First, a verification sequence minimizing the proxy is computed, then this sequence is refined by reintroducing the logical error rate. Finally, we identify a large group of automorphisms of the search space which preserve the proxy and we use this symmetry to reduce the size of the search space. This results in a 168 × (respectively 20,160 × ) reduction of the size of the search space for the optimization of verification sequences with three (respectively four) Pauli operators. Our numerical simulations for 20-qubit Clifford circuits with size 400 under the ion chain model show that our optimization algorithms improve the performance of CliNR by 25% and that the two-step optimization achieves the same results as the global optimization with 64% fewer evaluations of the logical error rate. Finally, we perform experiments on a 36-qubit trapped ion quantum computer, without mid-circuit measurements, showing that the CZNR variant of CliNR is at breakeven.
Clifford noise reduction (CliNR) is a partial error correction scheme that reduces the logical error rate of Clifford circuits at the cost of a modest qubit and gate overhead. The CliNR implementation of an
n-qubit Clifford circuit of size
s achieves a vanishing logical error rate if
snp2→0 where
p is the physical error rate. Here, we propose a recursive version of CliNR that can reduce errors on larger circuits with a relatively small gate overhead. When
np→0, the logical error rate can be vanishingly small. This implementation requires
(2⌈log(sp)⌉+3)n+1 qubits and at most $24 s \left\lceil(sp)^4\right\rceil $ gates. Using numerical simulations, we show that the recursive method can offer an advantage in a realistic near-term parameter regime. When circuit sizes are large enough, recursive CliNR can reach a lower logical error rate than the original CliNR with the same gate overhead. The results offer promise for reducing logical errors in large Clifford circuits with relatively small overheads.
Quantum simulation of fermionic Hamiltonians is a leading application of quantum computing, but accurate execution on present-day hardware is limited by error accumulation in deep Trotter circuits. We present a device-matched noise-reduction framework for encoded Hamiltonian simulation that combines symplectic-transvection-based Trotter synthesis in the Generalized Superfast Encoding (GSE) with Clifford Noise Reduction (CliNR) and Shor-style stabilizer verification enabled by mid-circuit measurement. We implement this approach for a six-qubit encoded Clifford Trotter step on a Barium development system similar to the forthcoming IonQ Tempo line and benchmark it against direct execution using both hardware experiments and a calibrated device-level noise model. The encoded CliNR execution achieves up to 54% lower logical error rate. Crucially, this advantage disappears when stabilizer readout is deferred to the end of the circuit, showing that timely mid-circuit fault detection, rather than verification overhead alone, drives the improvement. As a proof of concept, we further show that machine-learning-guided stabilizer selection can identify verification operators that outperform random choices. These results demonstrate that encoding-native verification combined with dynamic-circuit primitives can materially improve application-motivated quantum simulation without the full overhead of quantum error correction.
The preparation of high-fidelity logical magic states is a crucial subroutine for universal fault-tolerant quantum computation (FTQC). Predicting the performance of FTQC and developing improved protocols rely on numerical methods to classically simulate logical magic state preparation in the presence of noise. Clifford logic on Pauli-stabilizer codes with circuit-level Pauli errors can be efficiently simulated using Pauli-stabilizer formalism, but the non-Clifford operations required to prepare logical magic states render generic simulation inefficient. We introduce Clifford-stabilizer simulation, an exact and efficient algorithm based on updating a Clifford-stabilizer group to simulate noisy preparation protocols for a broad class of logical magic states used to implement non-Clifford gates in the third level of the Clifford hierarchy under circuit-level Pauli errors. Clifford-stabilizer simulation applies to a range of operations that commonly appear in preparation protocols for such logical magic states, including Pauli-stabilizer measurements, logical Clifford measurements, and transversal non-Clifford gates. Our algorithm for Clifford-stabilizer simulation maps a non-Clifford circuit with sampled circuit-level Pauli errors to a Clifford circuit that exactly reproduces its measurement outcome distribution, achieving time and space complexities polynomial in relevant protocol parameters. We perform exact simulation of magic state cultivation up to fault distance 7 by Clifford-stabilizer simulation. Our method provides a route to perform exact benchmarking of large-scale logical magic state preparation protocols required for useful FTQC.
We propose a Clifford noise reduction (CliNR) scheme that provides a reduction of the logical error rate of Clifford circuit with lower overhead than error correction and without the exponential sampling overhead of error mitigation. CliNR implements Clifford circuits by splitting them into subcircuits that are performed using gate teleportation. A few random stabilizer measurements are used to detect errors in the resources states consumed by the gate teleportation. This can be seen as a teleported version of the coherent parity-check scheme [1-3], with offline fault-detection making it scalable. We prove that CliNR achieves a vanishing logical error rate for families of n-qubit Clifford circuits with size s such that nsp^{2} goes to 0, where p is the physical error rate, meaning that it reaches the regime ns=o(1/p^{2}), whereas the direct implementation is limited to s=o(1/p). Moreover, CliNR uses only 3n+1 qubits, 2s+o(s) gates and has zero rejection rate. This small overhead makes it more practical than quantum error correction in the near term and our numerical simulations show that CliNR provides a reduction of the logical error rate in relevant noise regimes.
Practical fault-tolerant quantum computing requires qubit-efficient quantum error correction (QEC). Clifford deformations (CDs) modify the stabilizers of a QEC code to lower its logical error rate at no additional resource cost. However, the number of possible CDs grows exponentially with the number of qubits, making exhaustive search infeasible beyond small codes. In this work, we further show that under circuit level noise the optimal CD depends strongly on both the noise model and decoder. To make systematic CD optimization possible, we introduce Evolution-Based Best-arm Identification (EBBI), a shot-efficient search algorithm that treats the circuit, noise model and decoder as a black box. We benchmark EBBI's performance on the rotated surface code, under two strongly
Z-biased circuit-level noise models with the PyMatching decoder. At distance 3, EBBI finds the optimal CD using up to
∼1100× fewer shots than random search. At distance 5, when run at the target error rate, the median EBBI run converges to the CDs we believe to be optimal, while random search never converges. Under a
Z-biased SI1000-like noise model, EBBI discovers a new CD, the B-XZZX code, which we generalized to larger distances. It differs from XZZX on only a fraction
O(d−1) of the qubits at code distance
d, yet its relative advantage grows from
≈5% at
d=3 to
≈12% at
d=11. We thus show that the B-XZZX pattern found by EBBI can be systematically scaled to larger code distances. Our work represents a first step towards scalable CD optimization for general QEC codes.
Correcting errors in real time is essential for reliable large-scale quantum computations. Realizing this high-level function requires a system capable of several low-level primitives, including single-qubit and two-qubit operations, mid-circuit measurements of subsets of qubits, real-time processing of measurement outcomes, and the ability to condition subsequent gate operations on those measurements. In this work, we use a ten qubit QCCD trapped-ion quantum computer to encode a single logical qubit using the
[[7,1,3]] color code, first proposed by Steane~\cite{steane1996error}. The logical qubit is initialized into the eigenstates of three mutually unbiased bases using an encoding circuit, and we measure an average logical SPAM error of
1.7(6)×10−3, compared to the average physical SPAM error
2.4(8)×10−3 of our qubits. We then perform multiple syndrome measurements on the encoded qubit, using a real-time decoder to determine any necessary corrections that are done either as software updates to the Pauli frame or as physically applied gates. Moreover, these procedures are done repeatedly while maintaining coherence, demonstrating a dynamically protected logical qubit memory. Additionally, we demonstrate non-Clifford qubit operations by encoding a logical magic state with an error rate below the threshold required for magic state distillation. Finally, we present system-level simulations that allow us to identify key hardware upgrades that may enable the system to reach the pseudo-threshold.