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arXiv:1609.08188 · Journal of Mathematical Physics

Representations of the multi-qubit Clifford group

Jonas Helsen, Joel J. Wallman, Stephanie Wehner

Published Jul 1, 2018• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1063/1.4997688

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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.
The n-qubit Pauli group and its normalizer, the n-qubit Clifford group, have applications in quantum error correction and device characterization. Recent applications have used the representation theory of the Clifford group. We apply the tools of (the coincidentally named) Clifford theory to examine the representation theory of the Clifford group using the much simpler representation theory of the Pauli group. We find an unexpected correspondence between irreducible characters of the n-qubit Clifford group and those of the (n+1)-qubit Clifford group.
The Clifford group plays a central role in quantum information science. It is the building block for many error-correcting schemes and matches the first three moments of the Haar measure over the unitary group-a property that is essential for a broad range of quantum algorithms, with applications in pseudorandomness, learning theory, benchmarking, and entanglement distillation. At the heart of understanding many properties of the Clifford group lies the Clifford commutant: the set of operators that commute with k -fold tensor powers of Clifford unitaries. Previous understanding of this commutant has been limited to relatively small values of k , constrained by the number of qubits n . In this work, we develop a complete theory of the Clifford commutant. Our first result provides an explicit orthogonal basis for the commutant and computes its dimension for arbitrary n and k . We also introduce an alternative and easy-to-manipulate basis formed by isotropic sums of Pauli operators. We show that this basis is generated by products of permutations, which generate the unitary group commutant, and at most three other operators. Additionally, we develop a graphical calculus allowing a diagrammatic manipulation of elements of this basis. These results enable a wealth of applications: among others, we characterize all measurable magic measures and identify optimal strategies for stabilizer property testing, whose success probability also offers an operational interpretation to stabilizer entropies. Finally, we show that these results also generalize to multi-qudit systems with prime local dimension.
Based on a presentation of Cn and the help of [The GAP Group 2022, https://www.gap-system.org] we construct the character table of the Clifford group Cn for n = 1, 2, 3. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in Helsen et al. [J. Math. Phys. 59, 072201 (2018)] and Wallman and Flammia [New J. Phys. 16(10), 103032 (2014)] and reveal some new phenomena. We prove that when n ≥ 3, (1) the trivial character is the only linear character for Cn and hence Cn equals to its commutator subgroup, (2) the n-qubit Pauli group Pn is the only proper non-trivial normal subgroup of Cn, (3) the matrix representation M2n is a faithful representation for Cn. As a byproduct, we give a presentation of the finite symplectic group Sp(2n, 2) in terms of generators and relations.
NOTE: PAPER WITHDRAWN (See Comments) The Clifford and Local Clifford groups for d>2d > 2d>2 dimensional systems have been topics of recent interest due to their applications in graph states, quantum codes, and possible applications in fast quantum algorithms. This paper studies these groups more abstractly, by first characterizing the Pauli Automorphism and Local Automorphism groups, and then using these results to determine characteristics of the Clifford and Local Clifford groups. Not only does such an approach reveal new information about the Clifford and Local Clifford groups, but it also shows how many previously derived results arise naturally as simple corollaries. Lastly, we give a systematic method of building an arbitrary Local Clifford operator from a small number of previously known gates, as well as a method to physically implement such an operator.
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.
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.