Sdim: A Qudit Stabilizer Simulator

TL;DR

This work introduces Sdim, the first open-source stabilizer tableau simulator for qudits in prime dimensions, addressing a critical gap in numerical tools for qudit fault-tolerant quantum computing. Sdim represents states and circuit evolution via $O(n^2)$ tableaux and enables efficient sampling through Pauli frames, enabling large-scale, measurement-heavy benchmarking that outperforms state-vector methods in both reach and speed for entangled circuits. The authors validate correctness against Cirq and demonstrate practical utility through a qutrit Deutsch-Jozsa case study, Bernstein-Vazirani measurements, and a five-qutrit folded code benchmarking scenario, including multi-shot Pauli-frame sampling. They also discuss non-prime dimension challenges and outline a path toward broader FTQC infrastructure, highlighting significant implications for qudit QECC research and benchmarking workflows.

Abstract

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.

Figures (17)

• Figure 1: Average time to simulate one shot of a random Clifford circuit using our stabilizer versus Google Cirq statevector simulation. The space and time cost of state vector simulation grows exponentially versus the dimension and number of qudits. In contrast, tableau simulation does not depend on the dimension at all and is bounded by $n^2$.
• Figure 2: Conjugation table for Clifford gates on Pauli gates.
• Figure 3: A Sdim program to run and validate the $d$-dimensional Deutsch-Jozsa algorithm.
• Figure 4: The first three steps of the Deutsch-Jozsa algorithm written out in various representations, including the tableau.
• Figure 5: Tableau rewrite rules corresponding to the conjugation table in Figure \ref{['fig:conj-table']}.