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Physical Review A

Near-threshold qudit stabilizer codes with efficient encoding circuits for magic-state distillation

Abhi Kumar Sharma, Shayan Srinivasa Garani

Published Jun 18, 2024• 2 citations
Publisher

Abstract

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 ${[[31,1,16]]}_{2}$ quantum Bose-Chaudhuri-Hocquenghem code and ${[[29,1,15]]}_{3}$ 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.

DOI
10.1103/physreva.109.062426

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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⟩|T\ensuremath{\rangle}∣T\ensuremath⟩ and ∣S\ensuremath⟩|S\ensuremath{\rangle}∣S\ensuremath⟩ 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 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.
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.
We present applications of quantum quadratic residue codes in magic state distillation. This includes showing that existing codes which are known to distill magic states, like the 555-qubit perfect code, the 777-qubit Steane code, and the 111111-qutrit and 232323-qubit Golay codes, are equivalent to certain quantum quadratic residue codes. We also present new examples of quantum quadratic residue codes that distill qubit TTT states and qutrit Strange states with high thresholds, and we show that there are infinitely many quantum quadratic residue codes that distill TTT states with a non-trivial threshold. All of these codes, including the codes with the highest currently known thresholds for TTT state and Strange state distillation, are unified under the umbrella of quantum quadratic residue codes.
Producing high-fidelity magic states using the smallest possible number of physical qubits and operations stands as a very important challenge to achieve fault-tolerant quantum computation at scale. Besides emerging proposals for alternative methods such as cultivation, magic state distillation remains essential for achieving very low error rates. Known distillation protocols are usually built through quantum codes derived from triorthogonal matrices. Here, exploiting the specific noise structure present in magic state distillation protocols, we show that classical error-correcting codes offer a simpler framework for deriving these protocols. This formulation is particularly well suited to systematic numerical and analytical studies of distillation protocols involving a fixed number of qubits. Specifically, we use a SAT solver to derive a series of no-go theorems that relate key figures of merit, including the number of qubits and the factory distance. In particular, considering the standard implementation as a circuit of Z-diagonal Pauli product rotations followed by measurements, we show that no TTT-to-TTT distillation protocol on fewer than eight qubits can exceed distance 3, and no TTT-to-CCZ\mathrm{CC}ZCCZ protocol distance 2. Our results also include new such distillation protocols with the smallest number of qubits for a given distance in the literature, namely distance 4 and 5 TTT-to-TTT protocols supported on 10 and 11 qubits, as well as distance 3 and 4 TTT-to-CCZ\mathrm{CC}ZCCZ distillation protocols supported on 9 and 10 qubits. Finally, going beyond unitary circuits by recycling qubits through mid-circuit measurement and reinitialization, we obtain an implementation of the distance-5 49T49T49T-to-1T1T1T protocol on only 5 active qubits.
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.
Qudits offer the potential for low-overhead magic state distillation, although previous results for asymptotically good codes have required qudit dimension q≫100q\gg 100q≫100 or code length N≫100\mathcal{N}\gg 100N≫100. These parameters far exceed experimental demonstrations of qudit platforms, and thus motivate the search for better codes. Using a novel lifting procedure, we construct the first family of good triorthogonal codes on the F22m\mathbb{F}_{2^{2m}}F22m​ alphabet with m≥3m \geq 3m≥3 that lies above the Tsfasman-Vladut-Zink bound. These codes yield a family of asymptotically good quantum codes with transversal CCZ gates, enabling constant space overhead magic state distillation with qudit dimension as small as q=64q=64q=64. Further, we identify a promising code with parameters [[42,14,6]]64[[42,14,6]]_{64}[[42,14,6]]64​. Finally, we show that a distilled ∣CCZ⟩22m|{CCZ}\rangle_{2^{2m}}∣CCZ⟩22m​ can be reduced to a ∣CCZ⟩2n|{CCZ}\rangle_{2^n}∣CCZ⟩2n​ state for arbitrary nnn with a constant-depth Clifford circuit of at most 9 computational basis measurements, 12 single-qudit and 9 two-qudit Clifford gates.