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arXiv:1308.2493 · Physical Review A

Quantum circuits employing roots of the Pauli matrices

Mathias Soeken, D. Michael Miller, Rolf Drechsler

Published Oct 16, 2013• 41 citations
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Abstract

The Pauli matrices are a set of three $2\ifmmode\times\else\texttimes\fi{}2$ complex Hermitian unitary matrices. In this article, we investigate the relationships between certain roots of the Pauli matrices and how gates implementing those roots are used in quantum circuits. Techniques for simplifying such circuits are given. In particular, we show how those techniques can be used to find a circuit of $\text{Clifford}+T$ gates starting from a circuit composed of gates from the well-studied NOT, CNOT, V library.

DOI
10.1103/physreva.88.042322

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The Pauli matrices are a set of three 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 complex Hermitian unitary matrices. In this article, we investigate the relationships between certain roots of the Pauli matrices and how gates implementing those roots are used in quantum circuits. Techniques for simplifying such circuits are given. In particular, we show how those techniques can be used to find a circuit of Clifford+T\text{Clifford}+TClifford+T gates starting from a circuit composed of gates from the well-studied NOT, CNOT, V library.
The Clifford+T quantum computing gate library for single qubit gates can create all unitary matrices that are generated by the group 〈H, T〉. The matrix T can be considered the fourth root of Pauli Z, since T4= Z or also the eighth root of the identity I. The Hadamard matrix H can be used to translate between the Pauli matrices, since (HTH)4gives Pauli X. We are generalizing both these roots of the Pauli matrices (or roots of the identity) and translation matrices to investigate the groups they generate: the so-called Pauli root groups. In this work we introduce a formalization of such groups, study finiteness and infiniteness properties, and precisely determine equality and subgroup relations.
We examine the following problem: given a collection of Clifford gates, describe the set of unitaries generated by circuits composed of those gates. Specifically, we allow the standard circuit operations of composition and tensor product, as well as ancillary workspace qubits as long as they start and end in states uncorrelated with the input, which rule out common "magic state injection" techniques that make Clifford circuits universal. We show that there are exactly 57 classes of Clifford unitaries and present a full classification characterizing the gate sets which generate them. This is the first attempt at a quantum extension of the classification of reversible classical gates introduced by Aaronson et al., another part of an ambitious program to classify all quantum gate sets. The classification uses, at its center, a reinterpretation of the tableau representation of Clifford gates to give circuit decompositions, from which elementary generators can easily be extracted. The 57 different classes are generated in this way, 30 of which arise from the single-qubit subgroups of the Clifford group. At a high level, the remaining classes are arranged according to the bases they preserve. For instance, the CNOT gate preserves the X and Z bases because it maps X-basis elements to X-basis elements and Z-basis elements to Z-basis elements. The remaining classes are characterized by more subtle tableau invariants; for instance, the T_4 and phase gate generate a proper subclass of Z-preserving gates.
In fault-tolerant quantum computation and quan-tum error-correction one is interested on Pauli matrices that commute with a circuit/unitary. This information is encoded by the support (Pllaha et al., 2020) of the given circuit/unitary. We provide a fast algorithm that decomposes any Clifford gate as a minimal product of Clifford transvections. The algorithm can be directly used for computing the support of any given Clifford gate. To achieve this goal, we exploit the structure of the symplectic group with a novel graphical approach.
Optimizing T gates in Clifford+T circuit as π/4π/4π/4 rotations around Paulis
In this work, we introduce a new circuit optimization technique to reduce the number of T gates in Clifford+T circuits by treating T gates conjugated by Clifford gates as π4\fracπ{4}4π​-rotations around Pauli operators. The tested benchmarks shows up to 71.43%71.43\%71.43% and an average 42.67%42.67\%42.67% reduction in T-count, both surpass the best performance reported. The worst case complexity of our algorithm is O(nk2)O(nk^2)O(nk2) where nnn is the number of qubits and kkk is the number of T gates in the original Clifford+T circuit.
It is known that the matrices that can be exactly represented by a multiqubit circuit over the Toffoli+Hadamard, Clifford+TTT, or, more generally, Clifford-cyclotomic gate set are precisely the unitary matrices with entries in the ring Z[1/2,ζk]\mathbb{Z}[1/2,\zeta_k]Z[1/2,ζk​], where kkk is a positive integer that depends on the gate set and ζk\zeta_kζk​ is a primitive 2k2^k2k-th root of unity. In the present paper, we establish an analogous correspondence for qutrits. We define the multiqutrit Clifford-cyclotomic gate set of degree 3k3^k3k by extending the classical qutrit gates XXX, CXCXCX, and CCXCCXCCX with the Hadamard gate HHH and the TkT_kTk​ gate Tk=diag(1,ωk,ωk2)T_k=\mathrm{diag}(1,\omega_k, \omega_k^2)Tk​=diag(1,ωk​,ωk2​), where ωk\omega_kωk​ is a primitive 3k3^k3k-th root of unity. This gate set is equivalent to the qutrit Toffoli+Hadamard gate set when k=1k=1k=1, and to the qutrit Clifford+TkT_kTk​ gate set when k>1k>1k>1. We then prove that a 3n×3n3^n\times 3^n3n×3n unitary matrix UUU can be represented by an nnn-qutrit circuit over the Clifford-cyclotomic gate set of degree 3k3^k3k if and only if the entries of UUU lie in the ring Z[1/3,ωk]\mathbb{Z}[1/3,\omega_k]Z[1/3,ωk​].
IBM architectures impose some restrictions the quantum circuits that can be implemented. Only gates from Clifford+T gate library can be used and not all of the CNOT gates are available. Some of the CNOT need to be mapped into a sequence of gates. In this paper we present a set of mappings to conform to the restrictions imposed by IBM's architectures. These mappings require fewer gates than SWAP gates. It is well known, that permuting the qubits will yield circuits with different number of gates. The design in this paper uses efficient mappings with qubit permutations to obtain circuits with a reduced number of gates. Results have shown that the proposed approach reduces circuits by up to 64% compared with Qiskit and up to 42% compared with another mapping algorithm.