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International Conference on Communications

Magic State Distillation from Qudit Stabilizer Codes

Abhi Kumar Sharma, Shayan Srinivasa Garani

Published Jun 9, 2024• 0 citations
Publisher

Abstract

Magic state distillation (MSD) is a procedure to purify noisy non-stabilizer states from a quantum code, useful towards fault-tolerant quantum computing. Quantum circuits requiring non-Clifford gates can be realized using special states called magic states that are the eigenstates of both non-Clifford and Clifford gates. The purification of noisy states is realized by projecting these magic states onto the space spanned by the eigenstates of the projection operator of the stabilizer code. Given a noisy non- stabilizer state from a given stabilizer code, not all states can be distilled towards magic states. In this paper, we provide the condition for distillability of magic states specific to the choice of qudit stabilizer codes. We derive the noise threshold for magic state distillation depending on the code parameters. We prove that magic state distillation is a convergent map in the entropic sense. We also give an efficient decoding circuit for the best-known code to distill the$\left|H_i\right\rangle$state, which is an eigenstate of the qutrit Hadamard gate.

DOI
10.1109/icc51166.2024.10623071

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Magic state distillation (MSD) is a procedure to purify noisy non-stabilizer states from a quantum code, useful towards fault-tolerant quantum computing. Quantum circuits requiring non-Clifford gates can be realized using special states called magic states that are the eigenstates of both non-Clifford and Clifford gates. The purification of noisy states is realized by projecting these magic states onto the space spanned by the eigenstates of the projection operator of the stabilizer code. Given a noisy non- stabilizer state from a given stabilizer code, not all states can be distilled towards magic states. In this paper, we provide the condition for distillability of magic states specific to the choice of qudit stabilizer codes. We derive the noise threshold for magic state distillation depending on the code parameters. We prove that magic state distillation is a convergent map in the entropic sense. We also give an efficient decoding circuit for the best-known code to distill the∣Hi⟩\left|H_i\right\rangle∣Hi​⟩state, which is an eigenstate of the qutrit Hadamard gate.
Magic-state distillation is a protocol to purify noisy quantum states using quantum stabilizer codes. In this paper, we provide conditions for identifying magic states along with analytical expressions for computing the noise threshold. Through quantum entropic results, we prove the convergence of the magic-state distillation procedure. We derive analytical results to show how the noise threshold scales as a function of the minimum distance and the number of physical qudits of the underlying quantum stabilizer code. We distill $|T\ensuremath{\rangle}$ and $|S\ensuremath{\rangle}$ states with thresholds 0.498 (theoretical limit is 0.5) and 0.71 (theoretical limit is 0.75) over the depolarizing channel using [[31,1,16]]2{[[31,1,16]]}_{2}[[31,1,16]]2​ quantum Bose-Chaudhuri-Hocquenghem code and [[29,1,15]]3{[[29,1,15]]}_{3}[[29,1,15]]3​ quantum qutrit code, respectively. Finally, encoding algorithms based on quantum Calderbank-Shor-Steane (CSS) and non-CSS qudit stabilizer codes along with circuits are proposed for magic-state distillation.
Magic state distillation (MSD) is a purification protocol that plays an important role in fault-tolerant quantum computation. Repeated iteration of the steps of an MSD protocol generates pure single non-stabilizer states, or magic states, from multiple copies of a mixed resource state using stabilizer operations only. Thus mixed resource states promote the stabilizer operations to full universality. MSD was introduced for qubit-based quantum computation, but little has been known concerning MSD in higher-dimensional qudit-based computation. Here, we describe a general approach for studying MSD in higher dimensions. We use it to investigate the features of a qutrit MSD protocol based on the five-qutrit stabilizer code. We show that this protocol distils non-stabilizer magic states, and identify two types of states that are attractors of this iteration map. Finally, we show how these states may be converted, via stabilizer circuits alone, into a state suitable for state-injected implementation of a non-Clifford phase gate, enabling non-Clifford unitary computation.
Magic state distillation is a critical component in leading proposals for fault-tolerant quantum computation. Relatively little is known, however, about how to construct a magic state distillation routine or, more specifically, which stabilizer codes are suitable for the task. While transversality of a non-Clifford gate within a code often leads to efficient distillation routines, it appears to not be a necessary condition. Here we have examined a number of small stabilizer codes and highlight a handful of which displaying interesting, albeit inefficient, distillation behaviour. Many of these distill noisy states right up to the boundary of the known undististillable region, while some distill toward non-stabilizer states that have not previously been considered.
Magic State Distillation (MSD) has been a research focus for fault-tolerant quantum computing due to the need for non-Clifford resource in gaining quantum advantage. Although many of the MSD protocols so far are based on stabilizer codes with transversal T gates, there exists quite several protocols that don't fall into this class. Here we propose a method to map MSD protocols to iterative dynamical systems under the framework of stabilizer reduction. With the proposed mapping, we are able to analyze the performance of MSD protocols using techniques from dynamical systems theory, easily simulate the distillation process of input states under arbitrary noise and visualize it using flow diagram. We apply our mapping to common MSD protocols for |T⟩ state and find some interesting properties: The [[15,1,3]] code may distill states corresponding to T gate and the [[5,1,3]] code can distill the magic state corresponding to the T gate. Besides, we examine the exotic MSD protocols that may distill into other magic states proposed in [Eur. Phys. J. D 70, 55 (2016)] and identify the condition for distillable magic states. We also study new MSD protocols generated by concatenating different codes and numerically demonstrate that concatenation can generate MSD protocols with various magic states. By concatenating efficient codes with exotic codes, we can reduce the overhead of the exotic MSD protocols. We believe our proposed method will be a useful tool for simulating and visualization MSD protocols for canonical MSD protocols on |T⟩ as well as other unexplored MSD protocols for other states.
Determining the best attainable threshold for qudit magic state distillation is directly related to the question of whether or not contextuality is sufficient for universal quantum computation. We show that the performance of a qudit correcting code for magic state distillation is captured by its complete weight enumerator. For the qutrit strange state -- a maximally magic non-stabilizer state -- the performance of a code is captured by its simple weight enumerator. This result allows us to carry out an extensive search for high-threshold magic state distillation routines for the strange state. Our search covers all [[n,1]]3[[n,1]]_3[[n,1]]3​ qutrit stabilizer codes with a complete set of transversal Clifford gates for n≤23n\leq 23n≤23, and all [[n,1]]3[[n,1]]_3[[n,1]]3​ stabilizer codes with a transversal H2H^2H2 gate with n≤9n \leq 9n≤9 qudits. For n=23n=23n=23, we find over 600 CSS codes that can distill the qutrit strange state with cubic noise suppression. While none of these codes surpass the threshold of the 11-qutrit Golay code, their existence suggests that, for large codes, the ability to distill the qutrit strange state is somewhat generic.
Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely , using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill -type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately 100 × p 2 in terms of the physical error rate p . For example, with a physical error rate of p = 10 − 4 ( 10 − 3 ), the logical error rate is reduced to p L = 10 − 6 ( 10 − 4 ), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.