Unifying the Clifford Hierarchy via Symmetric Matrices over Rings
Narayanan Rengaswamy, Robert Calderbank, Henry D. Pfister
备注：
v3: Fixed Theorem 11. v2: Main correction - our framework does not include all diagonal gates but only all 2-local and certain higher-locality ones. All formulas still remain intact. v2 has more examples and remarks for clarity. Abstract updated. PDF marking diff to v1 is included in the package. v1: Presented as a poster in QIP 2019. RevTeX 4.2, 12 pages, two-column, 1 table. Examples included
The Clifford hierarchy is a foundational concept for universal quantum computation (UQC). It was introduced to show that UQC can be realized via quantum teleportation, given access to certain standard resources. While the full structure of the hierarchy is still not understood, Cui et al. (arXiv:1608.06596) recently described the structure of diagonal unitaries in the hierarchy. They considered diagonal gates whose action on a computational basis qudit state is described by a $2^k$-th root of unity raised to a polynomial function of the state, and they established the level of such unitaries in the hierarchy. For qubit systems, we consider $k$-th level diagonal gates that can be described just by quadratic forms of the state over the ring $\mathbb{Z}_{2^k}$ of integers mod $2^k$. These involve symmetric matrices over $\mathbb{Z}_{2^k}$ that can be used to efficiently describe all $2$-local and certain higher locality diagonal gates in the hierarchy. We also provide explicit algebraic descriptions of their action on Pauli matrices, which establishes a natural recursion to diagonal gates from lower levels. This involves symplectic matrices over $\mathbb{Z}_{2^k}$ and hence our perspective unifies these gates with the binary symplectic framework for Clifford gates. We augment our description with simple examples for certain standard gates. In addition to demonstrating structure, these formulas might prove useful in applications such as (i) classical simulation of quantum circuits, especially via the stabilizer rank approach, (ii) synthesis of logical non-Clifford unitaries, specifically alternatives to magic state distillation, and (iii) decomposition of arbitrary unitaries beyond the Clifford+$T$ set of gates, perhaps leading to shorter depth circuits. Our results suggest that some non-diagonal gates might be understood by generalizing other binary symplectic matrices to integer rings.
The Clifford hierarchy is a set of gates that appears in the theory of fault-tolerant quantum computation, but its precise structure remains elusive. We give a complete characterization of the diagonal gates in the Clifford hierarchy for prime-dimensional qudits. They turn out to be $p^{m}$-th roots of unity raised to polynomial functions of the basis state to which they are applied, and we determine which level of the Clifford hierarchy a given gate sits in based on $m$ and the degree of the polynomial.
We investigate the action of the first three levels of the Clifford hierarchy on sets of mutually unbiased bases comprising the Ivanovic MUB and the Alltop MUBs. Vectors in the Alltop MUBs exhibit additional symmetries when the dimension is a prime number equal to 1 modulo 3 and thus the set of all Alltop vectors splits into three Clifford orbits. These vectors form configurations with so-called Zauner subspaces, eigenspaces of order 3 elements of the Clifford group highly relevant to the SIC problem. We identify Alltop vectors as the magic states that appear in the context of fault-tolerant u…
Given oracle access to an unknown unitary C from the Clifford group and its conjugate, we give an exact algorithm for identifying C with O(n) queries, which we prove is optimal. We then extend this to all levels of the Gottesman-Chuang hierarchy (also known as the C_k hierarchy). Further, for unitaries not in the hierarchy itself but known to be close to an element of the hierarchy, we give a method of finding this close element. We also present a Clifford testing algorithm that decides whether a given black-box unitary is close to a Clifford or far from every Clifford.
Teleportation is a crucial element in fault-tolerant quantum computation and a complete understanding of its capacity is very important for the practical implementation of optimal fault-tolerant architectures. It is known that stabilizer codes support a natural set of gates that can be more easily implemented by teleportation than any other gates. These gates belong to the so called $\mathcal{C}_k$ hierarchy introduced by Gottesman and Chuang (Nature \textbf{402}, 390). Moreover, a subset of $\mathcal{C}_k$ gates, called semi-Clifford operations, can be implemented by an even simpler architect…
For universal quantum computation, a major challenge to overcome for practical implementation is the large amount of resources required for fault-tolerant quantum information processing. An important aspect is implementing arbitrary unitary operators built from logical gates within the quantum error correction code. A synthesis algorithm can be used to approximate any unitary gate up to arbitrary precision by assembling sequences of logical gates chosen from a small set of universal gates that are fault-tolerantly performable while encoded in a quantum error-correction code. However, current p…
In this article we investigate the possibility of encoding classical information onto multipartite quantum states in the distant laboratory framework. We show that for all states generated by Clifford operation there always exist such an encoding, this includes all stabilizer states such as cluster states and all graph states. We also show encoding for classes of symmetric states (which cannot be generated by Clifford operations). We generalise our approach using group theoretic methods introducing the unifying notion of Pseudo Clifford operations. All states generated by Pseudo Clifford opera…
Fault-tolerant quantum computation is a basic problem in quantum computation, and teleportation is one of the main techniques in this theory. Using teleportation on stabilizer codes, the most well-known quantum codes, Pauli gates and Clifford operators can be applied fault-tolerantly. Indeed, this technique can be generalized for an extended set of gates, the so called ${\mathcal{C}}_k$ hierarchy gates, introduced by Gottesman and Chuang (Nature, 402, 390-392). ${\mathcal{C}}_k$ gates are a generalization of Clifford operators, but our knowledge of these sets is not as rich as our knowledge of…
The teleportation model of quantum computation introduced by Gottesman and Chuang (1999) motivated the development of the Clifford hierarchy. Despite its intrinsic value for quantum computing, the widespread use of magic state distillation, which is closely related to this model, emphasizes the importance of comprehending the hierarchy. There is currently a limited understanding of the structure of this hierarchy, apart from the case of diagonal unitaries (Cui et al., 2017; Rengaswamy et al. 2019). We explore the structure of the second and third levels of the hierarchy, the first level being …
Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic
Institute for Advanced Study（高等研究院）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM T.J.沃森研究中心）；Kavli Institute of Theoretical Physics（卡弗里理论物理研究所）；King’s College London（伦敦国王学院）
We consider four-dimensional qudits as qubit pairs and their qudit Pauli operators as qubit Clifford operators. This introduces a nesting, $C_1^2 \subset C_2^4 \subset C_3^2$, where $C_n^m$ is the $n$th level of the $m$-dimensional qudit Clifford hierarchy. If we can convert between logical qubits and qudits, then qudit Clifford operators are qubit non-Clifford operators. Conversion is achieved by qubit fusion and qudit fission using stabilizer circuits that consume a resource state. This resource is a fused qubit stabilizer state with a fault-tolerant state preparation using stabilizer circui…
The design of a four-dimensional toric code is explored with the goal of finding a lattice capable of implementing a logical $\mathsf{CCCZ}$ gate transversally. The established lattice is the octaplex tessellation, which is a regular tessellation of four-dimensional Euclidean space whose underlying 4-cell is the octaplex, or hyper-diamond. This differs from the conventional 4D toric code lattice, based on the hypercubic tessellation, which is symmetric with respect to logical $X$ and $Z$ and only allows for the implementation of a transversal Clifford gate. This work further develops the estab…
The Clifford hierarchy is a nested sequence of sets of quantum gates critical to achieving fault-tolerant quantum computation. Diagonal gates of the Clifford hierarchy and 'nearly diagonal' semi-Clifford gates are particularly important: they admit efficient gate teleportation protocols that implement these gates with fewer ancillary quantum resources such as magic states. Despite the practical importance of these sets of gates, many questions about their structure remain open; this is especially true in the higher-dimensional qudit setting. Our contribution is to leverage the discrete Stone-v…
Clifford-hierarchy stabilizer formalism with applications to twisted quantum doubles
Joint Quantum Institute, NIST/University of Maryland（联合量子研究所，美国国家标准与技术研究院/马里兰大学）；Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（量子信息与计算机科学联合中心，美国国家标准与技术研究院/马里兰大学）；School of Physics, The University of Sydney（悉尼大学物理学院）；Institute for Physical Sciences and Technology, University of Maryland（马里兰大学物理科学与工程技术研究所）；Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland（马里兰大学物理系凝聚态理论中心与联合量子研究所）
该文提出 $X\mathcal C^{(k)}_d$ 稳定子形式体系，推广 Pauli 稳定子以涵盖扭曲量子双重模型，证明其可规避 Bravyi-Koenig 界，并展示通过测量该稳定子可确定性制备逻辑魔法态。
The leading paradigm for performing computation on quantum memories can be encapsulated as distill-then-synthesize. Initially, one performs several rounds of distillation to create high-fidelity magic states that provide one good T gate, an essential quantum logic gate. Subsequently, gate synthesis intersperses many T gates with Clifford gates to realise a desired circuit. We introduce a unified framework that implements one round of distillation and multi-qubit gate synthesis in a single step. Typically, our method uses the same number of T gates as conventional synthesis, but with the added …
Magic state distillation and the Shor factoring algorithm make essential use of logical diagonal gates. We introduce a method of synthesizing CSS codes that realize a target logical diagonal gate at some level $l$ in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of $Z$-stabilizers, and addition of $X$-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level $l$ inducing a logical diagonal gate …
Stabilizer codes are a simple and successful class of quantum error-correcting codes. Yet this success comes in spite of some harsh limitations on the ability of these codes to fault-tolerantly compute. Here we introduce a new metric for these codes, the disjointness, which, roughly speaking, is the number of mostly non-overlapping representatives of any given non-trivial logical Pauli operator. We use the disjointness to prove that transversal gates on error-detecting stabilizer codes are necessarily in a finite level of the Clifford hierarchy. We also apply our techniques to topological code…
University of Amsterdam（阿姆斯特丹大学）；QuSoft；Charles University, Prague, Czech Republic（布拉格查理大学）；Department of Computer Science, University of Oxford（牛津大学计算机系）
Clifford circuits -- i.e. circuits composed of only CNOT, Hadamard, and $π/4$ phase gates -- play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) -- where one additionally conjugates every qubit by the same one-qubit gate $U$ -- can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford c…
A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is $2^ν$ for a positive integer $ν$, then one can construct a Calderbank-Shor-Steane (CSS) code, where $X$-stabilizer space is the divisible classical code, that admits a transversal gate in the $ν$-th level of Clifford hierarchy. We consider a generalization of the divisibility by allowing a coefficient vector of odd integers with which every code word has zero dot product modulo the divisor. In this generalized sense, we construct a CSS code with divisor $2^{ν+1}$ and cod…
Parafermions are Zn generalisations of Majorana quasiparticles, with fractional non-Abelian statistics. They can be used to encode topological qudits and perform Clifford operations by their braiding. We study the simplest case of the Z3 parafermion chain and investigate the form of the non-topological gate that arises through direct short-range interaction of the parafermion edge modes. We show that such an interaction gives rise to a dynamical phase gate on the encoded ground space, with the strongest order of the interaction generating a non-Clifford gate which can be tuned to belong to eve…
We present a fault-tolerant universal gate set consisting of Hadamard and controlled-controlled-Z (CCZ) on Bacon-Shor subsystem codes. Transversal non-Clifford gates on these codes are intriguing in that higher levels of the Clifford hierarchy become accessible as the code becomes more asymmetric. For instance, in an appropriate gauge, Bacon-Shor codes on an $m\times m^k$ lattice have transversal $k$-qubit-controlled $Z$. Through a variety of tricks, including intermediate error-correction and non-Pauli recovery, we reduce the overhead required for fault-tolerant CCZ. We calculate pseudothresh…
When visualised as an operation on the Bloch sphere, the qubit "pi-over-eight" gate corresponds to one-eighth of a complete rotation about the vertical axis. This simple gate often plays an important role in quantum information theory, typically in situations for which Pauli and Clifford gates are insufficient. Most notably, when it supplements the set of Clifford gates then universal quantum computation can be achieved. The "pi-over-eight" gate is the simplest example of an operation from the third level of the Clifford hierarchy (i.e., it maps Pauli operations to Clifford operations under co…
