We develop a notion of quantum channels that can make states useless for universal quantum computation by destroying their magic (non-stabilizerness) - we refer to them as magic-breaking channels. We establish the properties of these channels in arbitrary dimensions. We prove the necessary and sufficient criteria for qubit channels to be magic-breaking and present an algorithm for determining the same. Moreover, we provide compact criteria in terms of the parameters for several classes of qubit channels to be magic-breaking under various post-processing operations. Further, we investigate the necessary and sufficient conditions for the tensor product of multiple qubit channels to be magic-breaking. We establish implications of the same for the dynamical resource theory of magic preservability.
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 …
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…
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.
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.
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…
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
Alice & Bob；Laboratoire de Physique de l’École Normale Supérieure, École Normale Supérieure, Centre Automatique et Systèmes, Mines Paris, Université PSL, CNRS, Inria（巴黎高等师范学院物理实验室，巴黎高等师范学院，自动与系统中心，巴黎矿业学院，PSL大学，法国国家科学研究中心，法国国家信息与自动化研究所）
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,…
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…
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.
Hardware-Tailored Resource Estimation for Magic-State Distillation on Silicon Spin Qubits
Cavendish Lab., Department of Physics, Univ. of Cambridge（剑桥大学卡文迪什实验室物理系）；Hitachi Cambridge Lab.（日立剑桥实验室）；Department of Computer Science and Technology, Univ. of Cambridge（剑桥大学计算机科学与技术系）
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
Silicon Quantum Computing Pty. Ltd.（硅量子计算有限公司）；UNSW Sydney（新南威尔士大学）；Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, UNSW Sydney（新南威尔士大学物理学院量子计算与通信技术卓越中心）
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…
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…
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…
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…
We consider the task of secure multi-party distributed quantum computation on a quantum network. We propose a protocol based on quantum error correction which reduces the number of necessary qubits. That is, each of the $n$ nodes in our protocol requires an operational workspace of $n^2+Θ(s) n$ qubits, as opposed to previously shown $Ω\big((n^3+n^2s^2)\log n\big)$ qubits, where $s$ is a security parameter. To achieve universal computation, we develop a distributed procedure for verifying magic states based on magic state distillation and statistical testing of randomly selected states. This al…
