Michael Zurel, Cihan Okay, Robert Raussendorf
We show that every quantum computation can be described by Bayesian update of a probability distribution on a finite state space. When applied to the model of quantum computation with magic states, the size of this state space only depends on the number of magic states used in the quantum computation, and not on the length of the gate and measurement sequence.
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For certain quantum architectures and algorithms, most of the required resources are consumed during the distillation of one-qubit magic states for use in performing Toffoli gates. I show that the overhead for magic-state distillation can be reduced by merging distillation with the implementation of Toffoli gates. The resulting routine distills 8 one-qubit magic states directly to a Toffoli state, which can be used without further magic to perform a Toffoli gate.
Magic sets of observables are minimal structures that capture quantum state-independent advantage for systems of $n\ge 2$ qubits and are, therefore, fundamental tools for investigating the interface between classical and quantum physics. A theorem by Arkhipov (arXiv:1209.3819) states that $n$-qubit magic sets in which each observable is in exactly two subsets of compatible observables can be reduced either to the two-qubit magic square or the three-qubit magic pentagram [N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990)]. An open question is whether there are magic sets that cannot be reduced to …
Qubit magic-breaking channels
本文定义并研究破魔量子比特信道，给出其充要判据、判定算法、多类信道参数条件及张量积条件，并阐明其对魔性动态资源理论的意义。
Hidden variable model for quantum computation with magic states on qudits of any dimension
本文将qubit上magic state通用量子计算的隐变量模型扩展至任意Hilbert空间维度的qudit，并导出经典模拟算法。
Magic state distillation with permutation-invariant codes and a two-qubit example
本文提出了一种基于排列不变码的双量子比特魔态蒸馏协议，实现了0.5误差阈值和1/2蒸馏率，通过允许非克利夫顿门和改变输出态形式，提升了魔态蒸馏性能。
Many proposals for fault-tolerant quantum computation require injection of 'magic states' to achieve a universal set of operations. Some qubit states are above a threshold fidelity, allowing them to be converted into magic states via 'magic state distillation', a process based on stabilizer codes from quantum error correction. We define quantum weight enumerators that take into account the sign of the stabilizer operators. These enumerators completely describe the magic state distillation behavior when distilling T-type magic states. While it is straightforward to calculate them directly by co…
The magic state injection process is a critical component of fault-tolerant quantum computing, and numerous studies have been conducted on this topic. Many existing studies have focused on square-lattice structures, where each qubit connects directly to four other qubits via two-qubit gates. However, hardware that does not follow a lattice structure, such as IBM's heavy-hexagon structure, is also under development. In these non-lattice structures, many quantum error correction (QEC) codes designed for lattice-based system cannot be directly applied. Adapting these codes often requires incorpor…
Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond
本文证明了单量子比特态在稳定子保真度、魔幻相对熵及α-z Rényi魔幻相对熵下的乘性或可加性，识别出连续可加魔幻单调性，并给出若干态的闭式表达式及魔幻态蒸馏开销的更紧下界。
Magic state distillation is one of the leading candidates for implementing universal fault-tolerant logical gates. However, the distillation circuits themselves are not fault-tolerant, so there is additional cost to first implement encoded Clifford gates with negligible error. In this paper we present a scheme to fault-tolerantly and directly prepare magic states using flag qubits. One of these schemes uses a single extra ancilla, even with noisy Clifford gates. We compare the physical qubit and gate cost of this scheme to the magic state distillation protocol of Meier, Eastin, and Knill, whic…
The distillation of magic states is an often-cited technique for enabling universal quantum computing once the error probability for a special subset of gates has been made negligible by other means. We present a routine for magic-state distillation that reduces the required overhead for a range of parameters of practical interest. Each iteration of the routine uses a four-qubit error-detecting code to distill the +1 eigenstate of the Hadamard gate at a cost of ten input states per two improved output states. Use of this routine in combination with the 15-to-1 distillation routine described by…
Magic state distillation uses special codes to suppress errors in input states, which are often tailored to a Clifford-twirled error model. We present detailed measurement sequences for magic state distillation protocols which can suppress arbitrary errors on any part of a protocol, assuming the independence of errors across qubits. Provided with input magic states, our protocol operates on a two-dimensional square grid by measurements of $ZZ$ on horizontal pairs of qubits, $XX$ on vertical pairs, and $Z,X$ on single qubits.
Constant-Overhead Magic State Injection into qLDPC Codes with Error Independence Guarantees
本文提出一种基于并行化码手术的通用可扩展方法，可将魔法态注入qLDPC码中任意逻辑量子比特，仅需恒定量子比特开销和$\ ilde{O}(d^2)$时间复杂度，并严格证明了注入魔法态的误差独立性，为降低容错量子计算资源需求提供新途径。
We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et …
Quantum resource theories are a powerful framework to characterize and quantify relevant quantum phenomena and identify processes that optimize their use for different tasks. Here, we define a resource measure for magic, the sought-after property in most fault-tolerant quantum computers. In contrast to previous literature, our formulation is based on bosonic codes, well-studied tools in continuous-variable quantum computation. Particularly, we use the Gottesman-Kitaev-Preskill code to represent multi-qubit states and consider the resource theory for the Wigner negativity. Our techniques are us…
Triangle Criterion: a mixed-state magic criterion with applications in distillation and detection
本文提出三角准则，用于衡量混合态魔性，类似正偏置转置准则用于纠缠。通过该准则揭示多量子比特魔性提纯和检测的新特性，并证明多量子比特提纯协议比单量子比特方案更强大。
A set of stabilizer operations augmented by some special initial states known as 'magic states', gives the possibility of universal fault-tolerant quantum computation. However, magic state preparation inevitably involves nonideal operations that introduce noise. The most common method to eliminate the noise is magic state distillation (MSD) by stabilizer operations. Here we propose a hybrid MSD protocol by connecting a four-qubit H-type MSD with a five-qubit T-type MSD, in order to overcome some disadvantages of the previous MSD protocols. The hybrid MSD protocol further integrates distillable…
The overhead cost of performing universal fault-tolerant quantum computation for large scale quantum algorithms is very high. Despite several attempts at alternative schemes, magic state distillation remains one of the most efficient schemes for simulating non-Clifford gates in a fault-tolerant way. However, since magic state distillation circuits are not fault-tolerant, all Clifford operations must be encoded in a large distance code in order to have comparable failure rates with the magic states being distilled. In this work, we introduce a new concept which we call redundant ancilla encodin…
The development of a framework for quantifying "non-stabiliserness" of quantum operations is motivated by the magic state model of fault-tolerant quantum computation, and by the need to estimate classical simulation cost for noisy intermediate-scale quantum (NISQ) devices. The robustness of magic was recently proposed as a well-behaved magic monotone for multi-qubit states and quantifies the simulation overhead of circuits composed of Clifford+T gates, or circuits using other gates from the Clifford hierarchy. Here we present a general theory of the "non-stabiliserness" of quantum operations r…
Unfolded distillation: very low-cost magic state preparation for biased-noise qubits
提出一种针对偏噪声量子比特的展开蒸馏方案，通过利用噪声偏置和展开Hadamard 3D量子Reed-Muller码的X稳定子群，在物理层而非逻辑层实现魔法态蒸馏，显著降低电路体积并保持高保真度。
Magic State Injection with Erasure Qubits
通过擦除量子比特提高魔法态注入的逻辑错误率控制，减少空间-时间开销，适用于表面码和颜色码的容错量子计算。
Constant-Overhead Magic State Distillation
本文提出利用代数几何码构造量子比特魔法态蒸馏协议，实现恒定开销（指数γ=0），优于此前最佳γ≈0.678，解决了最优指数蒸馏的开放问题。
We demonstrate high fidelity single-qubit gate operation in a trapped single neutral atom. The atom is trapped in the recently invented magic-intensity optical dipole trap (MI-ODT) with more stable magnetic field. The MI-ODT efficiently mitigates the detrimental effects of light shifts thus sufficiently improves the performance of single qubit-gates. The gates are driven with microwave, and the fidelity of gate operation is characterized by using the randomized benchmarking method. We obtain an average error per Clifford gate of $3.0(7)\times10^{-5}$ which is much below the error threshold ($1…
We describe a scheme of quantum computation with magic states on qubits for which contextuality is a necessary resource possessed by the magic states. More generally, we establish contextuality as a necessary resource for all schemes of quantum computation with magic states on qubits that satisfy three simple postulates. Furthermore, we identify stringent consistency conditions on such computational schemes, revealing the general structure by which negativity of Wigner functions, hardness of classical simulation of the computation, and contextuality are connected.
Despite significant overhead reductions since its first proposal, magic state distillation is often considered to be a very costly procedure that dominates the resource cost of fault-tolerant quantum computers. The goal of this work is to demonstrate that this is not true. By writing distillation circuits in a form that separates qubits that are capable of error detection from those that are not, most logical qubits used for distillation can be encoded at a very low code distance. This significantly reduces the space-time cost of distillation, as well as the number of qubits. In extreme cases,…
High-threshold magic state distillation with quantum quadratic residue codes
本文探讨了量子二次剩余码在魔态提纯中的应用，展示了现有提纯魔态的编码与量子二次剩余码等价，并提出新的高阈值提纯编码。
In-Situ Simultaneous Magic State Injection on Arbitrary CSS qLDPC Codes
本文提出一种原位魔态注入方案，可在任意CSS qLDPC码中直接生成逻辑魔态，利用仅需提取 syndrome 的资源，通过模拟展示其在不同噪声模型下的优越性。
We present a simple and precise protocol for standard quantum teleportation of N-qubit state, considering the most general resource q-channel and Bell states. We find condition on these states for perfect teleportation and give explicitly the unitary transformation required to be done by Bob for achieving perfect teleportation. We discuss connection of our simple theory with the complicated related work on this subject and with character matrix, transformation, judgment and kernel operators defined in this context. We also prove that the magic basis discussed by Hill and Wootters [Phys. Rev. L…
Quantum algorithms for quantum chemistry map the dynamics of electrons in a molecule to the dynamics of a coupled spin system. To reach chemical accuracy for interesting molecules, a large number of quantum gates must be applied which implies the need for quantum error correction and fault-tolerant quantum computation. Arbitrary fault-tolerant operations can be constructed from a small, universal set of fault-tolerant operations by gate compilation. Quantum chemistry algorithms are compiled by decomposing the dynamics of the coupled spin-system using a Trotter formula, synthesizing the decompo…
Distilling Magic States in the Bicycle Architecture
本文提出在双变量自行车码中实现基于泡利测量的克利福德电路，通过联合优化逻辑量子比特映射、门调度、测量本地化和量子比特回收来设计蒸馏工厂，在更小量子比特占用下实现更低目标错误率。
Eigenstates of permutation gates are either stabilizer states (for gates in the Pauli group) or magic states, thus allowing universal quantum computation [M. Planat and Rukhsan-Ul-Haq, Preprint 1701.06443]. We show in this paper that a subset of such magic states, when acting on the generalized Pauli group, define (asymmetric) informationally complete POVMs. Such IC-POVMs, investigated in dimensions $2$ to $12$, exhibit simple finite geometries in their projector products and, for dimensions $4$ and $8$ and $9$, relate to two-qubit, three-qubit and two-qutrit contextuality.
Generation of entanglement and magic via continuous homodyne monitoring of a qubit pair
本研究通过连续零差监测两个量子比特，利用碰撞模型推导随机主方程，证明该监测能产生纠缠并增强量子魔力，且资源可通过光学相位调节。
The relative entropy of magic and its nonadditivity
本文研究了魔法态的相对熵，并证明了其在大多数情况下非可加性。
Hardware-Tailored Resource Estimation for Magic-State Distillation on Silicon Spin Qubits
针对硅自旋量子比特平台，通过结合自底向上和自顶向下的方法，分析多种架构和纠错码，评估魔法态蒸馏的物理资源需求，并展示优化控制脉冲和偏置纠错码可显著降低开销。
Transversal Gates and Magic State Distillation in an Optimally Synthesized Spin-Qubit Shuttling Bus
本研究提出多量子比特自旋量子比特穿梭总线架构，优化布局与辅助量子比特共享方案，结合Quantum Reverse Mapping方法，实现容错量子计算的层间协同设计，验证了自旋量子比特穿梭架构的可行性。
Exploring the landscape of compact magic-state distillation factories
本文利用经典纠错码框架研究魔法态蒸馏协议，通过SAT求解器推导出关于量子比特数、协议深度、工厂距离和输出错误率预因子的不可行定理，并提出了新的最小量子比特数蒸馏协议。
A non-Clifford gate is required for universal quantum computation, and, typically, this is the most error-prone and resource intensive logical operation on an error-correcting code. Small, single-qubit rotations are popular choices for this non-Clifford gate, but certain three-qubit gates, such as Toffoli or controlled-controlled-Z (CCZ), are equivalent options that are also more suited for implementing some quantum algorithms, for instance those with coherent classical subroutines. Here, we calculate error rates and resource overheads for implementing logical CCZ with pieceable fault-toleranc…
To date all self-tests for high dimensional systems are confined to many-qubit states. This is due to two restrictions in the literature: the standard techniques for proving self-testing results apply only to qubits, and there are few suitable non-local games that are known to require high dimensional non-qubit states to achieve their optimal quantum value. In this paper we address these two problems. Specifically, we generalize the usual self-testing framework to q$d$its and then apply it to a generalization of the magic square game, giving a self-test for two maximally entangled pairs of q$d…
In this work, we construct a polarization-mediated magic-intensity (MI) optical dipole trap (ODT) array, in which the detrimental effects of light shifts on the mixed-species qubits are efficiently mitigated so that the coherence times of the mixed-species qubits are both substantially enhanced and balanced for the first time. This mixed-species magic trapping technique relies on the tunability of the coefficient of the third-order cross term and ground state hyperpolarizability, which are inherently dependent on the degree of circular polarization of the trap laser. Experimentally, polarizati…
The Magic Scroll: Leveraging biased noise to improve magic state cultivation in register-based architectures
本研究提出名为“魔法卷轴”的方法，结合偏置噪声与高量子比特连接性，改进魔法态制备，提升制备与蒸馏体积，降低错误率，适用于基于寄存器的容错量子计算架构。
Basis-independent stabilizerness and maximally noisy magic states
本文研究了绝对稳定子态和绝对威金纳正态态的特征，揭示了非稳定子态和威金纳负态的纯度下限，并提出了基于纯度的必要条件。
Generation of steady quantumness in the presence of an environment is of utmost importance if we are to build practical quantum devices. We propose a scheme of generating steady coherence and magic in a qubit system attached to a heat bath at a certain temperature through its interaction with another qubit system attached to a spin bath. Coherence generation in the reduced qubit is always possible in this model. The steady coherence in the reduced qubit attached to the heat bath may be used to enhance the subsequent transient performance of a quantum absorption refrigerator. For the case of ge…
The leading approach to fault tolerant quantum computing requires a continual supply of magic states. When a new magic state is first encoded, its initial fidelity will be too poor for use in the computation. This necessitates a resource-intensive distillation process that occupies the majority of the computer's hardware; creating magic states with a high initial fidelity minimises this cost and is therefore crucial for practical quantum computing. Here we present the surprising and encouraging result that raw magic states can have a fidelity significantly better than that of the two-qubit gat…
Sudden death of entanglement, rebirth of magic
研究探讨了局部马尔可夫噪声对纠缠和魔法的影响，发现纠缠无法恢复但魔法可以，揭示了魔法与纠缠之间的互补性。
Magic State Distillation via Codes over Binary Extension Fields
本研究采用代数几何技术构造二元扩展域上的码，提出多量子比特门封装方案与蒸馏协议，其在多数场景下的性能优于现有技术。
A Phase-Space Geometric Measure of Magic in Qubit Systems
本文提出一种基于相空间几何的量子魔性度量方法，通过研究稳定子Wigner函数的凸包与离散Wigner函数的l1距离，揭示了稳定子扩展与相空间几何之间的关系，并证明了该度量在量子纠错中的应用。
We present an infinite family of protocols to distill magic states for $T$-gates that has a low space overhead and uses an asymptotic number of input magic states to achieve a given target error that is conjectured to be optimal. The space overhead, defined as the ratio between the physical qubits to the number of output magic states, is asymptotically constant, while both the number of input magic states used per output state and the $T$-gate depth of the circuit scale linearly in the logarithm of the target error $δ$ (up to $\log \log 1/δ$). Unlike other distillation protocols, this protocol…
We demonstrate that the coherence of a single mobile atomic qubit can be well preserved during a transfer process among different optical dipole traps (ODTs). This is a prerequisite step in realizing a large-scale neutral atom quantum information processing platform. A qubit encoded in the hyperfine manifold of $^{87}$Rb atom is dynamically extracted from the static quantum register by an auxiliary moving ODT and reinserted into the static ODT. Previous experiments were limited by decoherences induced by the differential light shifts of qubit states. Here we apply a magic-intensity trapping te…
Given stabilizer operations and the ability to repeatedly prepare a single-qubit mixed state rho, can we do universal quantum computation? As motivation for this question, "magic state" distillation procedures can reduce the general fault-tolerance problem to that of performing fault-tolerant stabilizer circuits. We improve the procedures of Bravyi and Kitaev in the Hadamard "magic" direction of the Bloch sphere to achieve a sharp threshold between those rho allowing universal quantum computation, and those for which any calculation can be efficiently classically simulated. As a corollary, the…
Low Overhead Qutrit Magic State Distillation
本研究构建了适用于魔态蒸馏的三正交qutrit纠错码族，证明其可大幅降低容错量子计算的魔态蒸馏开销，小参数下的码性能已优于同规模的已知qubit三正交码。
Stabilizer Tensor Networks with Magic State Injection
本文提出魔态注入增强的 STN 框架，将少量 T 门随机 Clifford 电路模拟降为多项式复杂度，并在隐藏位移位基准上高效模拟 4000 量子比特、320 个 T 门。
The fermionic linear optical extent is multiplicative for 4 qubit parity eigenstates
本文研究了费米子线性光学（FLO）广度的标度行为，证明了任意纯态与4量子比特宇称本征态张量积的FLO广度等于两因子广度之积，并递归证明了该广度对4量子比特魔态的任意张量积具有可乘性。
We propose families of protocols for magic state distillation -- important components of fault tolerance schemes --- for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic state distillation scheme ye…
Encoding a magic state with beyond break-even fidelity
本研究提出并实现了基于超导量子比特阵列的纠错魔态制备方案，所得魔态质量优于单量子比特制备结果，还通过自适应电路提升了产率，验证了容错量子计算原理，可减少大规模架构中高保真魔态制备的物理比特需求。
The Pareto Frontiers of Magic and Entanglement: The Case of Two Qubits
研究双量子比特系统中魔法与纠缠的相互作用，确定最大魔法和最小魔法在给定纠缠水平下的帕累托前沿，并推导出分析公式。
Low-overhead Magic State Circuits with Transversal CNOTs
针对横向CNOT实验进展下的容错计算魔态制备问题，本文通过可重编译简化连续多量子比特相位旋转电路的算法，构建了更低开销的CCZ、CS、T态容错电路，为相关容错计算提供支撑。
Experimental Demonstration of High-Fidelity Logical Magic States from Code Switching
该研究在离子阱量子处理器上首次实验演示了色码间的码切换，制备出高保真逻辑魔态，其不保真度低于物理操作，为通用容错量子计算提供了关键原语。
LiDMaS: Architecture-Level Modeling of Fault-Tolerant Magic-State Injection in GKP Photonic Qubits
LiDMaS通过架构级建模研究GKP编码光子量子比特中逻辑T门魔态的容错注入，结合RUS协议和外表面码保护，提供可扩展光子容错架构的设计指导。
Magic state distillation is a crucial component in the leading approaches to implementing universal fault tolerant quantum computation, with existing protocols for both qubit and higher dimensional systems. Early work focused on determining the region of distillable states for qubit protocols, yet comparatively little is known about which states can be distilled and with what distillable region for d>2. Here we focus on d=3 and present new four-qutrit distillation schemes that improve upon the known distillable region, and achieve distillation tight to the boundary of undistillable states for …
In this paper we study reusable magic states. These states are a special subset of the standard magic states. Once distilled, reusable magic states can be used, repeatedly, to apply some unitary U. Given this property, reusable magic states have the potential to greatly lower qubit and gate overheads in fault-tolerant quantum computation. While these states are promising, we provide a strong argument for their limited computational power. Specifically, we show that if reusable magic states can be used to apply non-Clifford unitaries, then we can exploit them to efficiently simulate poly-sized …
Magic (non-stabilizerness) is a necessary but "expensive" kind of "fuel" to drive universal fault-tolerant quantum computation. To properly study and characterize the origin of quantum "complexity" in computation as well as physics, it is crucial to develop a rigorous understanding of the quantification of magic. Previous studies of magic mostly focused on small systems and largely relied on the discrete Wigner formalism (which is only well behaved in odd prime power dimensions). Here we present an initiatory study of the magic of genuinely many-body quantum states that may be strongly entangl…
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