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arXiv:2408.07764

Constant-Overhead Magic State Distillation

Adam Wills, Min-Hsiu Hsieh, Hayata Yamasaki

Published Aug 14, 2024• 4 citations
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Abstract

Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol's overhead, defined as the number of input magic states required per output magic state with an error rate below $ε$, typically grows as $\mathcal{O}(\log^γ(1/ε))$. Achieving smaller overheads, i.e., smaller exponents $γ$, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some $γ> 0$, and identifying the smallest achievable exponent $γ$ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an $\mathcal{O}(1)$ overhead, meaning the optimal exponent $γ= 0$; this improves over the previous best of $γ\approx 0.678$ due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for $2^{10}$-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. The use of asymptotically good codes with non-vanishing rate and relative distance leads to the constant overhead. These codes can be realised by representing each $2^{10}$-dimensional qudit as a set of $10$ qubits, using stabiliser operations on qubits. The $10$-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-$Z$ ($CCZ$) and $T$ gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent.

DOI
10.48550/arxiv.2408.07764

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Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum computation. The protocol's overhead, defined as the number of input magic states required per output magic state with an error rate below εεε, typically grows as O(log⁡γ(1/ε))\mathcal{O}(\log^γ(1/ε))O(logγ(1/ε)). Achieving smaller overheads, i.e., smaller exponents γγγ, is highly desirable; however, all existing protocols require polylogarithmically growing overheads with some γ>0γ> 0γ>0, and identifying the smallest achievable exponent γγγ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an O(1)\mathcal{O}(1)O(1) overhead, meaning the optimal exponent γ=0γ= 0γ=0; this improves over the previous best of γ≈0.678γ\approx 0.678γ≈0.678 due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for 2102^{10}210-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. The use of asymptotically good codes with non-vanishing rate and relative distance leads to the constant overhead. These codes can be realised by representing each 2102^{10}210-dimensional qudit as a set of 101010 qubits, using stabiliser operations on qubits. The 101010-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-ZZZ (CCZCCZCCZ) and TTT gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distillation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent.
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.
Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty T measurement, achieving infidelities of O ( p 3 ) for physical noise strength p . To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately 2 × 10 − 16 at p = 10 − 3 ), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.
Zero-level CCZCCZCCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the TTT gate and the CCZCCZCCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZCCZCCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T†T/T^\daggerT/T† operation of the [[8,3,2]][[ 8,3,2 ]][[8,3,2]] code to fault-tolerantly encode the state CCZ‾∣+++⟩\overline{CCZ}|+++\rangleCCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2p_L \simeq 300 \times p^2pL​≃300×p2 with respect to the physical error rate ppp. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3p = 10^{-3}p=10−3 and p=10−4p = 10^{-4}p=10−4, respectively, compared to conventional seven-TTT-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZCCZCCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional TTT-state generation.
The overhead exponent -- characterizing the scaling of the number of noisy magic states with respect to the target distillation error -- has been a central quantity to benchmark magic state distillation protocols. On the other hand, a related but less investigated quantity motivated by an information-theoretic viewpoint is the asymptotic distillation rate, the largest ratio of output to input magic states such that error vanishes asymptotically. These two quantities are tightly related in the specific case -- the overhead exponent is zero if and only if the asymptotic distillation rate is linear. However, their relationship in other regimes has been unclear. Here, we show that their quantitative relation is generally not robust, by presenting a family of magic state distillation protocols with an overhead exponent not close to zero -- in fact, larger than one -- that still achieves the asymptotic rate arbitrarily close to the linear rate. This implies that the distillation rate is not constrained by the overhead exponent within the sublinear rate regime. Notably, our protocol is based on error checking by measurements of logical Clifford operators, which underlies the recent magic state cultivation protocol, suggesting the potential of this mechanism for asymptotic magic state distillation.
Magic state distillation is a leading but costly approach to fault-tolerant quantum computation, and it is important to explore all possible ways of minimizing its overhead cost. The number of ancillae required to produce a magic state within a target error rate εεε is O(log⁡γ(ε−1))O(\log^γ (ε^{-1}))O(logγ(ε−1)) where γγγ is known as the yield parameter. Hastings and Haah derived a family of distillation protocols with sublogarithmic overhead (i.e., γ<1γ< 1γ<1) based on punctured Reed-Muller codes. Building on work by Campbell \textit{et al.} and Krishna-Tillich, which suggests that qudits of dimension p>2p>2p>2 can significantly reduce overhead, we generalize their construction to qudits of arbitrary prime dimension ppp. We find that, in an analytically tractable puncturing scheme, the number of qudits required to achieve sublogarithmic overhead decreases drastically as ppp increases, and the asymptotic yield parameter approaches 1ln⁡p\frac{1}{\ln p}lnp1​ as p→∞p \to \inftyp→∞. We also perform a small computational search for optimal puncture locations, which results in several interesting triorthogonal codes, including a [[519,106,5]]5[[519,106,5]]_5[[519,106,5]]5​ code with γ=0.99γ=0.99γ=0.99.
In order to circumvent that no universal set of gates can be achieved on a given quantum error-correcting codes, additional gates can be performed through magic state injection. This however requires to have access to high fidelity magic state, which can be obtained by the distillation of low quality states to fewer higher quality ones. In this paper, we provide a family of distillation protocols able to produce diagonal states at every level of the Clifford hierarchy. This family is obtained recursively using code doubling techniques and leveraging qubit recycling, the fact that some idle qubits of the protocols can be measured and re-used. At a given level of the Clifford hierarchy, we show that the protocols we derive can be performed at arbitrary high distance on a fix number of logical qubits. This family recovers many known efficient protocols, and uncovers new ones, such as a 111→1111 \to 1111→1 protocol for ∣T⟩|T\rangle∣T⟩ state distillation at distance 777, which are the most compact in term of volume. Finally, we discuss possible extension of this framework for distillation protocols producing more than one output state.