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arXiv:1510.02619 · Physical Review A

Multiqubit Clifford groups are unitary 3-designs

Huangjun Zhu

Published Dec 29, 2017• 182 citations
arXivHTMLPDF

Abstract

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 $t\ensuremath{\ge}3$ 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 in 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 in studying quantum computation.

DOI
10.1103/physreva.96.062336

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Unitary ttt-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 ttt-designs with t\ensuremath≥3t\ensuremath{\ge}3t\ensuremath≥3 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 in 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 in studying quantum computation.
A unitary t-design is a set of unitaries that is "evenly distributed" in the sense that the average of any t-th order polynomial over the design equals the average over the entire unitary group. In various fields -- e.g. quantum information theory -- one frequently encounters constructions that rely on matrices drawn uniformly at random from the unitary group. Often, it suffices to sample these matrices from a unitary t-design, for sufficiently high t. This results in more explicit, derandomized constructions. The most prominent unitary t-design considered in quantum information is the multi-qubit Clifford group. It is known to be a unitary 3-design, but, unfortunately, not a 4-design. Here, we give a simple, explicit characterization of the way in which the Clifford group fails to constitute a 4-design. Our results show that for various applications in quantum information theory and in the theory of convex signal recovery, Clifford orbits perform almost as well as those of true 4-designs. Technically, it turns out that in a precise sense, the 4th tensor power of the Clifford group affords only one more invariant subspace than the 4th tensor power of the unitary group. That additional subspace is a stabilizer code -- a structure extensively studied in the field of quantum error correction codes. The action of the Clifford group on this stabilizer code can be decomposed explicitly into previously known irreps of the discrete symplectic group. We give various constructions of exact complex projective 4-designs or approximate 4-designs of arbitrarily high precision from Clifford orbits. Building on results from coding theory, we give strong evidence suggesting that these orbits actually constitute complex projective 5-designs.
Unitary t-designs are some of the most versatile tools in quantum information theory. Their applications range from randomized benchmarking and shadow tomography, to more fundamental ones such as emulating quantum chaos and establishing exponential separations between classical and quantum query complexity. While unitary designs originating from a group structure, such as the Clifford group, have proven to be incredibly useful for qubit systems, unfortunately, this is no longer true for qudits. In fact, the classification of finite-group representations rules out the existence of unitary 2-designs for arbitrary qudit dimensions. This severely limits the applicability of standard quantum information primitives when it comes to qudit systems. We overcome these limitations with a three-fold contribution. First, we introduce a general technique to construct families of weighted state t-designs in arbitrary qudit dimensions. These weighted state-designs generalize classical shadow tomography protocol from qubits to qudits. Second, we introduce a Clifford character RB that allows us to benchmark the qudit Clifford group in any dimension, including non-prime-power dimensions. And third, we establish bounds on the quantum circuit complexity of generating approximate unitary-designs from native gates in existing quantum hardware such as high-spin and cavity-QED qudits. Our work further highlights the analogy between spin and optical coherent states by proving that spin-GKP codewords form a state 2-design while spin coherent states do not; in direct analogy with the optical case. This work is structured as a pedagogical and self-contained introduction to unitary designs and their applications to qudit systems.
The Wigner function provides a useful quasiprobability representation of quantum mechanics, with applications in various branches of physics. Many nice properties of the Wigner function are intimately connected with the high symmetry of the underlying operator basis composed of phase point operators: any pair of phase point operators can be transformed to any other pair by a unitary symmetry transformation. We prove that, in the discrete scenario, this permutation symmetry is equivalent to the symmetry group being a unitary 2 design. Such a highly symmetric representation can only appear in odd prime power dimensions besides dimensions 2 and 8. It suffices to single out a unique discrete Wigner function among all possible quasiprobability representations. In the course of our study, we show that this discrete Wigner function is uniquely determined by Clifford covariance, while no Wigner function is Clifford covariant in any even prime power dimension.
Unitary k-designs are finite ensembles of unitary matrices that approximate the Haar distribution over unitary matrices. Several ensembles are known to be 2-designs, including the uniform distribution over the Clifford group, but no family of ensembles was previously known to form a 3-design. We prove that the Clifford group is a 3-design, showing that it is a better approximation to Haar-random unitaries than previously expected. Our proof strategy works for any distribution of unitaries satisfying a property we call Pauli 2-mixing and proceeds without the use of heavy mathematical machinery. We also show that the Clifford group does not form a 4-design, thus characterizing how well random Clifford elements approximate Haar-random unitaries. Additionally, we show that the generalized Clifford group for qudits is not a 3-design unless the dimension of the qudit is a power of 2.
We have generalized the well-known statement that the Clifford group is a unitary 3-design into symmetric cases by extending the notion of unitary design.Concretely, we have proven that a symmetric Clifford group is a symmetric unitary 3-design if and only if the symmetry constraint is described by some Pauli subgroup.We have also found a complete and unique construction method of symmetric Clifford groups with simple quantum gates for Pauli symmetries.For the overall understanding, we have also considered physically relevant U(1) and SU(2) symmetry constraints, which cannot be described by a Pauli subgroup, and have proven that the symmetric Clifford group is a symmetric unitary 1-design but not a 2-design under those symmetries.Our findings are numerically verified by computing the frame potentials, which measure the difference in randomness between the uniform ensemble on the symmetric group of interest and the symmetric unitary group.This work will open a new perspective into quantum information processing, such as randomized benchmarking, and give a deep understanding to many-body systems, such as monitored random circuits.
We give an explicit treatment of the third tensor power of the Clifford representation on nnn qutrits, for n≥2n\geq2n≥2. This representation has five pairwise inequivalent irreducible constituents, with multiplicities 1,1,1,1,21,1,1,1,21,1,1,1,2. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective 333-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.