Quantum computers have steadily improved over the last decade, but developing fault-tolerant quantum computing (FTQC) techniques, required for useful, universal computation remains an ongoing effort. Key elements of FTQC such as error-correcting codes and decoding are supported by a rich bed of stabilizer simulation software such as Stim and CHP, which are essential for numerically characterizing these protocols at realistic scales. Recently, experimental groups have built nascent high-dimensional quantum hardware, known as qudits, which have a myriad of attractive properties for algorithms and FTQC. Despite this, there are no widely available qudit stabilizer simulators. We introduce the first open-source realization of such a simulator for all dimensions. We demonstrate its correctness against existing state vector simulations and benchmark its performance in evaluating and sampling quantum circuits. This simulator is the essential computational infrastructure to explore novel qudit error correction as earlier stabilizer simulators have been for qubits.
Quantum computers have steadily improved over the last decade, but developing fault-tolerant quantum computing (FTQC) techniques, required for useful, universal computation remains an ongoing effort. Key elements of FTQC such as error-correcting codes and decoding are supported by a rich bed of stabilizer simulation software such as Stim and CHP, which are essential for numerically characterizing these protocols at realistic scales. Recently, experimental groups have built nascent high-dimensional quantum hardware, known as qudits, which have a myriad of attractive properties for algorithms and FTQC. Despite this, there are no widely available qudit stabilizer simulators. We introduce the first open-source realization of such a simulator for all dimensions. We demonstrate its correctness against existing state vector simulations and benchmark its performance in evaluating and sampling quantum circuits. This simulator is the essential computational infrastructure to explore novel qudit error correction as earlier stabilizer simulators have been for qubits.
We demonstrate that Pauli frames, a mathematical object used to efficiently sample qubit stabilizer circuits, can be used to efficiently sample qudit circuits of any prime dimension. Quantum error correcting codes are a central object of study in fault-tolerance quantum computing, which is necessary to realize useful universal quantum computing. Stabilizer codes belong to a large class of well-understood quantum codes that are crucially efficiently simulable on classical hardware using the so-called tableau representation. Popular simulation software like Stim exclusively studies and characterizes qubit codes, which entails repeatedly sampling a noisy quantum circuit with Pauli frames. We highlight the key differences (and fundamental similarities) between qubit and qudit stabilizer tableaus. It then teaches how Pauli frames generate batches of samples given a qudit tableau via two guided examples: one with a random measurement outcome but no noise, and one with a single noise gate.
Qudits can be described by a state vector in a q-dimensional Hilbert space, enabling a more extensive encoding and manipulation of information compared to qubits. This implies that conducting fault-tolerant quantum computations using qudits rather than qubits might entail less overhead. In this work, we investigate the viability of qudits in error correction codes by creating and simulating the quantum circuitry for the smallest qudit error correction code with a multidimensional circuit-level noise model and specifically adapted decoders. After introducing a flag qudit to protect the code from hook errors, comparable error thresholds of the order of 10−4 are obtained for qudits of dimensions 2, 3 and 5.
Qudits can be described by a state vector in a q-dimensional Hilbert space, enabling a more extensive encoding and manipulation of information compared to qubits. This implies that conducting fault-tolerant quantum computations using qudits rather than qubits might entail less overhead. In this work, I investigate the viability of qudits across three key components of a fault-tolerant quantum computer: hardware control, quantum error-correction codes, and logical gate operations. This includes benchmarking physically decomposed qudit gates against qubit-based implementations for a specific donor-spin system, threshold simulations of qudit error-correction codes using tailored simulators and decoders, and simulations of magic state distillation circuits with qudits. At the hardware level, a numerical leakage-aware randomized benchmarking protocol is applied to a native ququart encoding in a donor-spin system and compared against an equivalent pair of encoded qubits. The ququart implementation is found to reduce the lower-bound gate infidelity by 40 – 50%, a consequence of the fewer ESR and EDSR pulses and correspondingly lower cumulative noise exposure. This provides a concrete, platform-specific demonstration of reduced physical-gate overhead prior to the introduction of any error-correction redundancy. At the level of quantum error correction, the viability of qudits is investigated through the construction and simulation of the quantum circuits for the five-qudit perfect code. The code is evaluated under a multidimensional circuit-level noise model using decoders specifically adapted to the qudit setting. This yields the first threshold estimates reported for qutrit and ququint codes under circuit-level noise. After introducing a flag qudit to protect the code from hook errors, comparable error thresholds of the order of 10–4 are obtained for qudits of dimensions 2, 3 and 5. These simulations were subsequently performed using Sdim, a stabilizer tableau simulator that enabled threshold estimation up to dimension q = 7, a regime inaccessible to state-vector simulators such as Cirq. At the level of logical operations, magic state distillation protocols of the Bravyi-Kitaev type are simulated under logical circuit-level noise, to my knowledge for the first time for this protocol family. Once the Clifford operations implementing the distillation circuit are subject to a nonzero logical error rate, the achievable output fidelity saturates within a finite usable distillation range rather than an arbitrarily improvable one: roughly 0.9-15.6% for the |T⟩₂-type protocol, 3.7-5% for the |H⟩₂-type protocol and 5.4-7.4% for the |H+⟩₃-type protocol at a representative logical circuit-level noise level of pCL = 10–3. The qutrit protocol achieves a more favorable distillation threshold but suffers from a substantially lower yield due to its reduced syndrome acceptance probability. This highlights a trade-off between threshold performance and yield that is intrinsic to the higher-dimensional construction. Further investigation with more advanced and larger-scale protocols is required to better understand and characterize this behavior. Overall, these results show that qudits can provide real efficiency benefits: higher-dimensional control can improve physical gate fidelity, and qutrit and ququint error-correction thresholds can match those of qubits when hook errors are properly handled. However, logical circuit-level noise limits the maximum magic state fidelity, showing that the benefits of qudits are not yet fully realized at the logical-operation level. This thesis provides the first broad numerical study of these effects and introduces the simulation tools needed to explore larger code families, decoders, and distillation protocols.
Learning a classical description of a quantum state is a fundamental task in quantum computation. Among the most important classes of quantum states are stabilizer states, which play a central role in quantum error correction and fault-tolerant computation. To mitigate the effects of realistic noise, agnostic learning of stabilizer states has emerged as a natural and well-motivated problem. Recently, Chen \textit{et al.} [STOC'25, p. 429-438] resolved this problem for qubit systems by using a stabilizer bootstrapping framework. However, the agnostic learning of qudit stabilizer states remains largely unexplored, since the qudit setting introduces fundamental structural differences that prevent a direct generalization of existing qubit techniques. In this paper, we successfully generalize the stabilizer bootstrapping framework to qudit systems and present the first efficient quantum algorithm for agnostic learning of qudit stabilizer states. Specifically, given copies of an unknown n-qudit pure state ∣ψ⟩ that has fidelity τ with some stabilizer state, our algorithm outputs a stabilizer state ∣φ⟩ such that ∣⟨φ∣ψ⟩∣2≥τ−ε with high probability. The algorithm uses only single-copy and four-copy measurements, and its sample and time complexity scale as (d/τ)O(d2log(1/τ))⋅poly(n,1/ε), where the dimension d is an odd prime. As a direct corollary, our algorithm enables efficient estimation of the magic of a quantum state, as quantified by its stabilizer fidelity. Completing the picture, we also present a streamlined algorithm for the high-fidelity regime τ>cos2(π/8), establishing a qudit analogue of the threshold-based approach in prior qubit work.
As the field of quantum computing moves towards the realization of Quantum Error Correction (QEC), an increasing amount of attention has been paid to the implementation of decoders-classical systems analyzing the state of error on the quantum device in real-time. However, today there is a lack of tools available to characterize and compare the performance of decoders. In this work, we address this need and introduce decoder-bench, a framework for benchmarking decoders on relevant QEC code traces. decoder-bench integrates with Stim, a high performance tool for analyzing stabilizer circuits, to create traces for a variety of QEC codes and fault-tolerant subroutines. We use decoder-bench to evaluate the accuracy and latency performance of multiple decoders for circuit-level simulations of color code memory experiments, bivariate-bicycle code memory experiments, surface code memory experiments, and surface code lattice surgery experiments.
This paper presents “Stim", a fast simulator for quantum stabilizer circuits. The paper explains how Stim works and compares it to existing tools. With no foreknowledge, Stim can analyze a distance 100 surface code circuit (20 thousand qubits, 8 million gates, 1 million measurements) in 15 seconds and then begin sampling full circuit shots at a rate of 1 kHz. Stim uses a stabilizer tableau representation, similar to Aaronson and Gottesman's CHP simulator, but with three main improvements. First, Stim improves the asymptotic complexity of deterministic measurement from quadratic to linear by tracking the i n v e r s e of the circuit's stabilizer tableau. Second, Stim improves the constant factors of the algorithm by using a cache-friendly data layout and 256 bit wide SIMD instructions. Third, Stim only uses expensive stabilizer tableau simulation to create an initial reference sample. Further samples are collected in bulk by using that sample as a reference for batches of Pauli frames propagating through the circuit.