Clifford Group Research Articles

RETRACTED ARTICLE: Noise tailoring for quantum circuits via unitary 2t-design

Because of environmental variations and imperfect operations, real-world quantum computers produce different coherent errors that are difficult to estimate. Here, we propose a method whereby the twirled noise over a unitary 2t-design (a set of unitary matrices that approximate the entire unitary group) for quantum circuits can be tailored into stochastic noise. Then, we prove that local random circuits for twirling separable noisy channel over the Clifford group can be used to construct a unitary 2t-design, which is easy to implement in experiments. Moreover, we prove that our method is robust to gate-dependent and gate-independent noise. The stochastic noise can be both estimated by average fidelity and directly obtained by randomized benchmarking via unitary 2t-designs. Obtaining such tailored noise is an important guarantee for achieving fault-tolerant quantum computation.

SICs and the elements of order three in the Clifford group

For over a decade, there has been intensive work on the numerical and analytic construction of SICs (d2 equiangular lines in ) as an orbit of the Heisenberg group. The Clifford group, which consists of the unitary matrices which normalise the Heisenberg group, plays a key role in these constructions. All of the known fiducial (generating) vectors for such SICs are eigenvectors of symplectic operations in the Clifford group with canonical order 3. Here we describe the Clifford group and the subgroup of symplectic operations in terms of a natural set of generators. From this, we classify all its elements of canonical order three. In particular, we show (contrary to prior claims) that there are symplectic operations of canonical order 3 for , which are not conjugate to the Zauner matrix. It is as yet unknown whether these give rise to SICs.

Towards Large-scale Functional Verification of Universal Quantum Circuits

We introduce a framework for the formal specification and verification of quantum circuits based on the Feynman path integral. Our formalism, built around exponential sums of polynomial functions, provides a structured and natural way of specifying quantum operations, particularly for quantum implementations of classical functions. Verification of circuits over all levels of the Clifford hierarchy with respect to either a specification or reference circuit is enabled by a novel rewrite system for exponential sums with free variables. Our algorithm is further shown to give a polynomial-time decision procedure for checking the equivalence of Clifford group circuits. We evaluate our methods by performing automated verification of optimized Clifford+T circuits with up to 100 qubits and thousands of T gates, as well as the functional verification of quantum algorithms using hundreds of qubits. Our experiments culminate in the automated verification of the Hidden Shift algorithm for a class of Boolean functions in a fraction of the time it has taken recent algorithms to simulate.

Spectral Data for Spin Higgs Bundles

Abstract In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and “twisted” spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program, which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case).

Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes

One of the most promising routes towards fault-tolerant quantum computation utilizes topological quantum error correcting codes, such as the $\mathbb{Z}_2$ surface code. Logical qubits can be encoded in a variety of ways in the surface code, based on either boundary defects, holes, or bulk twist defects. However proposed fault-tolerant implementations of the Clifford group in these schemes are limited and often require unnecessary overhead. For example, the Clifford phase gate in certain planar and hole encodings has been proposed to be implemented using costly state injection and distillation protocols. In this paper, we show that within any encoding scheme for the logical qubits, we can fault-tolerantly implement the full Clifford group by using joint measurements involving a single appropriately encoded logical ancilla. This allows us to provide new low overhead implementations of the full Clifford group in surface and color codes. It also provides the first proposed implementations of the full Clifford group in hyperbolic codes. We further use our methods to propose state-of-the art encoding schemes for small numbers of logical qubits; for example, for code distances $d = 3,5,7$, we propose a scheme using $60, 160, 308$ (respectively) physical data qubits, which allow for the full logical Clifford group to be implemented on two logical qubits. To our knowledge, this is the optimal proposal to date, and thus may be useful for demonstration of fault-tolerant logical gates in small near-term quantum computers.

Quantum Security Computation on Shared Secrets

Ouyang et al. proposed an $(n,n)$ threshold quantum secret sharing scheme, where the number of participants is limited to $n=4k+1,k\in Z^+$, and the security evaluation of the scheme was carried out accordingly. In this paper, we propose an $(n,n)$ threshold quantum secret sharing scheme for the number of participants $n$ in any case ( $n\in Z^+$ ). The scheme is based on a quantum circuit, which consists of Clifford group gates and Toffoli gate. We study the properties of the quantum circuit in this paper and use the quantum circuit to analyze the security of the scheme for dishonest participants.

Construction of color codes from polygons

We propose a procedure for constructing new classes of codes, called polygonal color codes. This technique expands the triangular color codes introduced by Bombin and Martin-Delgado in 2007. Triangular color codes, built on a two-dimensional Euclidean surface with three edges, implement the entire Clifford group, but they only encode one qubit. In our case, polygonal color codes are constructed on two-dimensional surfaces with n edges, with n ≥ 3 , increasing the number of encoded qubits. Considering a surface with boundary and evaluating the generators of the first homology group of this surface, we may increase the dimension of the code without losing the transversal implementation of the Clifford group. In addition, we construct two codes, one with 2 and one with 3 encoded qubits, which exemplify the method and show that we can generate codes with better parameters than the known triangular codes. For one of these constructions, we also give general parameters.

Approximate randomized benchmarking for finite groups

We investigate randomized benchmarking (RB) in a general setting with quantum gates that form a representation, not necessarily an irreducible one, of a finite group. We derive an estimate for the average fidelity, to which experimental data may then be calibrated. Furthermore, we establish that RB can be achieved by the sole implementation of quantum gates that generate the group as well as one additional arbitrary group element. In this case, we need to assume that the noise is close to being covariant. This yields a more practical approach to RB. Moreover, we show that RB is stable with respect to approximate Haar sampling for the sequences of gates. This opens up the possibility of using Markov chain Monte Carlo methods to obtain the random sequences of gates more efficiently. We demonstrate these results numerically using the well-studied example of the Clifford group as well as the group of monomial unitary matrices. For the latter, we focus on the subgroup with nonzero entries consisting of nth roots of unity, which contains T gates.

Real Randomized Benchmarking

Randomized benchmarking provides a tool for obtaining precise quantitative estimates of the average error rate of a physical quantum channel. Here we define real randomized benchmarking, which enables a separate determination of the average error rate in the real and complex parts of the channel. This provides more fine-grained information about average error rates with approximately the same cost as the standard protocol. The protocol requires only averaging over the real Clifford group, a subgroup of the full complex Clifford group, and makes use of the fact that it forms an orthogonal 2-design. It therefore allows benchmarking of fault-tolerant gates for an encoding which does not contain the full Clifford group transversally. Furthermore, our results are especially useful when considering quantum computations on rebits (or real encodings of complex computations), in which case the real Clifford group now plays the role of the complex Clifford group when studying stabilizer circuits.

On Clifford groups in quantum computing

The term Clifford group was introduced in 1998 by D. Gottesmann in his investigation of quantum error-correcting codes. The simplest Clifford group in multiqubit quantum computation is generated by a restricted set of unitary Clifford gates - the Hadamard, π/4-phase and controlled-X gates. Because of this restriction the Clifford model of quantum computation can be efficiently simulated on a classical computer (the Gottesmann-Knill theorem). However, this fact does not diminish the importance of the Clifford model, since it may serve as a suitable starting point for a full-fledged quantum computation.In the general case of a single or composite quantum system with finite-dimensional Hilbert space the finite Weyl-Heisenberg group of unitary operators defines the quantum kinematics and the states of the quantum register. Then the corresponding Clifford group is defined as the group of unitary operators leaving the Weyl-Heisenberg group invariant. The aim of this contribution is to show that our comprehensive results on symmetries of the Pauli gradings of quantum operator algebras - covering any single as well as composite finite quantum systems - directly correspond to Clifford groups defined as quotients with respect to U(1).

Representations of the multi-qubit Clifford group

The q-qubit Clifford group, that is, the normalizer of the q-qubit Pauli group in U(2q), is a fundamental structure in quantum information with a wide variety of applications. We characterize all irreducible subrepresentations of the two-copy representation φ⊗2 of the Clifford group on the two-fold tensor product of the space of linear operators M2q⊗2. In the companion paper [Helsen et al., e-print arXiv:1701.04299 (2017)], we apply this result to improve the statistics of randomized benchmarking, a method for characterizing quantum systems.

Randomized benchmarking with restricted gate sets

Standard randomized benchmarking protocols entail sampling from a unitary 2 design, which is not always practical. In this article we examine randomized benchmarking protocols based on subgroups of the Clifford group that are not unitary 2 designs. We introduce a general method for analyzing such protocols and subsequently apply it to two subgroups, the group generated by controlled-NOT, Hadamard, and Pauli gates and that generated by only controlled-NOT and Pauli gates. In both cases the error probability can be estimated to within a factor of two or less where the factor can be arranged to be conservative and to decay exponentially in the number of qubits. For randomized benchmarking of logical qubits even better accuracy will typically be obtained. Thus, we show that sampling a distribution which is close to a unitary 2 design, although sufficient, is not necessary for randomized benchmarking to high accuracy.

Qutrit ZX-calculus is Complete for Stabilizer Quantum Mechanics

In this paper, we show that a qutrit version of ZX-calculus, with rules significantly different from that of the qubit version, is complete for pure qutrit stabilizer quantum mechanics, where state preparations and measurements are based on the three dimensional computational basis, and unitary operations are required to be in the generalized Clifford group. This means that any equation of diagrams that holds true under the standard interpretation in Hilbert spaces can be derived diagrammatically. In contrast to the qubit case, the situation here is more complicated due to the richer structure of this qutrit ZX-calculus.

A note on optimality of quantum circuits over metaplectic basis

Metaplectic quantum basis is a universal multi-qutrit quantum basis, formed by the ternary Clifford group and the axial reflection gate R = |0ih0| + |1ih1| − |2ih2|. It is arguably, a ternary basis with the simplest geometry. Recently Cui, Kliuchnikov, Wang and the Author have proposed a compilation algorithm to approximate any twolevel Householder reflection to precision ε by a metaplectic circuit of R-count at most C log3 (1/ε) + O(log log 1/ε) with C = 8. A new result in this note takes the constant down to C = 5 for non-exceptional target reflections under a certain credible numbertheoretical conjecture. The new method increases the chances of obtaining a truly optimal circuit but may not guarantee the true optimality. Efficient approximations of an important ternary quantum gate proposed by Howard, Campbell and others is also discussed.

Spark deficient Gabor frames

The theory of Gabor frames of functions defined on finite abelian groups was initially developed in order to better understand the properties of Gabor frames of functions defined over the reals. However, during the last twenty years the topic has acquired an interest of its own. One of the fundamental questions asked in this finite setting is the existence of full spark Gabor frames. The author proved the existence, as well as constructed such frames, when the underlying group is finite cyclic. In this paper, we resolve the non-cyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian non-cyclic groups. We also prove that all eigenvectors of certain unitary matrices in the Clifford group in odd dimensions generate spark deficient Gabor frames. Finally, similarities between the uncertainty principles concerning the finite dimensional Fourier transform and the short-time Fourier transform are discussed.

Multiqubit Clifford groups are unitary 3-designs

Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.

Discrete Wigner formalism for qubits and noncontextuality of Clifford gates on qubit stabilizer states

We show that qubit stabilizer states can be represented by non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive state-independent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two---producing an exterior, or Grassmann, algebra---which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each stabilizer state's three-generator Wigner function, and these distributions evolve deterministically to one another under Clifford gates. This corresponds to a hidden variable theory that is non-contextual and local for qubit Clifford gates while Clifford (Pauli) measurements have a context-dependent representation. Equivalently, we show that qubit Clifford gates can be expressed as propagators within the three-generator Wigner-Weyl-Moyal formalism whose semiclassical expansion is truncated at order $\hbar^0$ with a finite number of terms. The $T$-gate, which extends the Clifford gate set to one capable of universal quantum computation, require a semiclassical expansion of the propagator to order $\hbar^1$. We compare this approach to previous quasi-probability descriptions of qubits that relied on the two-generator Wigner-Weyl-Moyal formalism and find that the two-generator Weyl symbols of stabilizer states result in a description of evolution under Clifford gates that is state-dependent, in contrast to the three-generator formalism. We have thus extended Wigner non-negative quasi-probability distributions from the odd $d$-dimensional case to $d=2$ qubits, which describe the non-contextuality of Clifford gates and contextuality of Pauli measurements on qubit stabilizer states.

Semiclassical formulation of the Gottesman-Knill theorem and universal quantum computation

We give a path-integral formulation of the time evolution of qudits of odd dimension. This allows us to consider semiclassical evolution of discrete systems in terms of an expansion of the propagator in powers of $\ensuremath{\hbar}$. The largest power of $\ensuremath{\hbar}$ required to describe the evolution is a traditional measure of classicality. We show that the action of the Clifford operators on stabilizer states can be fully described by a single contribution of a path integral truncated at order ${\ensuremath{\hbar}}^{0}$ and so are ``classical,'' just like propagation of Gaussians under harmonic Hamiltonians in the continuous case. Such operations have no dependence on phase or quantum interference. Conversely, we show that supplementing the Clifford group with gates necessary for universal quantum computation results in a propagator consisting of a finite number of semiclassical path-integral contributions truncated at order ${\ensuremath{\hbar}}^{1}$, a number that nevertheless scales exponentially with the number of qudits. The same sum in continuous systems has an infinite number of terms at order ${\ensuremath{\hbar}}^{1}$.

Poking Holes and Cutting Corners to Achieve Clifford Gates with the Surface Code

The surface code is currently the leading proposal to achieve fault-tolerant quantum computation. Among its strengths are the plethora of known ways in which fault-tolerant Clifford operations can be performed, namely, by deforming the topology of the surface, by the fusion and splitting of codes and even by braiding engineered Majorana modes using twist defects. Here we present a unified framework to describe these methods, which can be used to better compare different schemes, and to facilitate the design of hybrid schemes. Our unification includes the identification of twist defects with the corners of the planar code. This identification enables us to perform single-qubit Clifford gates by exchanging the corners of the planar code via code deformation. We analyse ways in which different schemes can be combined, and propose a new logical encoding. We also show how all of the Clifford gates can be implemented with the planar code without loss of distance using code deformations, thus offering an attractive alternative to ancilla-mediated schemes to complete the Clifford group with lattice surgery.

The surface code with a twist

The surface code is one of the most successful approaches to topological quantum error-correction. It boasts the smallest known syndrome extraction circuits and correspondingly largest thresholds. Defect-based logical encodings of a new variety called twists have made it possible to implement the full Clifford group without state distillation. Here we investigate a patch-based encoding involving a modified twist. In our modified formulation, the resulting codes, called triangle codes for the shape of their planar layout, have only weight-four checks and relatively simple syndrome extraction circuits that maintain a high, near surface-code-level threshold. They also use 25% fewer physical qubits per logical qubit than the surface code. Moreover, benefiting from the twist, we can implement all Clifford gates by lattice surgery without the need for state distillation. By a surgical transformation to the surface code, we also develop a scheme of doing all Clifford gates on surface code patches in an atypical planar layout, though with less qubit efficiency than the triangle code. Finally, we remark that logical qubits encoded in triangle codes are naturally amenable to logical tomography, and the smallest triangle code can demonstrate high-pseudothreshold fault-tolerance to depolarizing noise using just 13 physical qubits.