We develop a simple and systematic route to higher levels of the qubit Clifford hierarchy by coherently controlling Clifford operations. Our approach is based on Pauli periodicity, defined for a Clifford unitary $U$ as the smallest integer $m\ge 1$ such that $U^{2^{m}}$ is a Pauli operator up to phase. We prove a sharp controlled-jump rule showing that the controlled gate $CU$ lies strictly in level $m+2$ of the hierarchy, and equivalently that $CU$ lies in level $k$ if $U^{2^{k-2}}$ is Pauli while no smaller positive power of $U$ is Pauli. We further quantify the resources required to realize large level jumps in the Clifford hierarchy by proving an essentially tight upper bound on Pauli periodicity as a function of the number of qubits, which implies that accessing high hierarchy levels through controlled Cliffords requires a number of target qubits that grows exponentially with the desired level. We complement this limitation with explicit infinite families of Pauli-periodic Cliffords whose controlled versions achieve asymptotically optimal jumps. As an application, we propose a protocol for preparing logical catalyst states that enable logical $Z^{1/2^k}$ phase gates via phase kickback from a single jumped Clifford.
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.
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…
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）
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 i…
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…
Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Mobius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimen…
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。
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 …
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…
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）
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…
Two-level quantum systems, qubits, are not the only basis for quantum computation. Advantages exist in using qudits, d-level quantum systems, as the basic carrier of quantum information. We show that color codes, a class of topological quantum codes with remarkable transversality properties, can be generalized to the qudit paradigm. In recent developments it was found that in three spatial dimensions a qubit color code can support a transversal non-Clifford gate, and that in higher spatial dimensions additional non-Clifford gates can be found, saturating Bravyi and König's bound [Phys. Rev. Le…
Fault-Tolerant Non-Clifford GKP Gates using Polynomial Phase Gates and On-Demand Noise Biasing
QuTech, Delft University of Technology（代尔夫特理工大学 QuTech）；Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学 卡弗里纳米科学研究所）；Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学 应用数学研究所）
We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form $\exp(-i\fracπ{2^{n}} Z\otimes Z)$. When $n = 2$, these are equivalent, up to local Clifford unitaries, to graph states. However, when $n > 2$, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an importan…
We investigate the geometric phase of a two-level atom (qubit) coupled to a bosonic reservoir with Lorentzian spectral density, and find that for the non-Markovian dynamics in which rotating-wave approximation (RWA) is performed, geometric phase has a $π$-phase jump at the nodal point. However, the exact result without RWA given by hierarchy equation of motion method shows that there is no such a phase jump or nodal structure in geometric phase. Thus our results demonstrate that the counter-rotating terms significantly contribute to the geometric phase in multi-mode Hamiltonian under certain c…
Hypergraph states are generalizations of graph states where controlled-$Z$ gates on edges are replaced with generalized controlled-$Z$ gates on hyperedges. Hypergraph states have several advantages over graph states. For example, certain hypergraph states, such as the Union Jack states, are universal resource states for measurement-based quantum computing with only Pauli measurements, while graph state measurement-based quantum computing needs non-Clifford basis measurements. Furthermore, it is impossible to classically efficiently sample measurement results on hypergraph states with a constan…
