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arXiv:1503.08579 · IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Translating Between the Roots of the Identity in Quantum Computers

Wouter Castryck, Jeroen Demeyer, Alexis De Vos, Oliver Keszöcze, Mathias Soeken

Published May 1, 2018• 2 citations
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Abstract

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.

DOI
10.1109/ismvl.2018.00051

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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.
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 Clifford+T\text{Clifford}+TClifford+T gates starting from a circuit composed of gates from the well-studied NOT, CNOT, V library.
We give three finite presentations in terms of generators and relations for three groups of operators. The operators are frequently used in quantum computation. The result can potentially be used to optimize quantum circuits. Two of the three proofs use the Reidemeister-Schreier theorem. Both the Reidemeister-Schreier theorem and the two proofs using it have been formally verified in the proof assistant Agda. First, we give a finite presentation of the group of 2-qubit Clifford+T operators. The proof applies the Reidemeister-Schreier theorem to a finite presentation of a supergroup of the 2-qubit Clifford+T group that was given by Greylyn. The process generates hundreds of relations. We then simplify them to 20 relations using a Pauli rotation representation as an almost-normal form. Second, we give a finite presentation of U_n (Z[1/2, i]), the group of unitary n × n- matrices with entries in Z[1/2, i]. Amy, Glaudell, and Ross showed that this group is generated by a certain set of two-level matrices, and they also gave an exact synthesis algorithm that converts any element of U_n (Z[1/2, i]) to a word in these generators. We say that a word is in normal form if it is an output of the exact synthesis algorithm. Using what we call semantically guided rewriting, we show that each word in the two-level generators is equivalent to a normal form under the congruence generated by finitely many relations. Third, we give a finite presentation of the group of 3-qubit Clifford+CS operators. Amy, Glaudell, and Ross proved that up to a condition on the determinant, the n-qubit Clifford+CS group is exactly U_(2^n) (Z[1/2, i]). In particular, the 3-qubit Clifford+CS group is a subgroup of U_8 (Z[1/2, i]) of index 2. Using a similar technique as in our first result, we repeatedly apply the Reidemeister-Schreier theorem to the presentation of U_8 (Z[1/2, i]) (from our second result), and then simplify thousands of relations into 17 relations. To do the simplification, we devise an almost-normal form for Clifford+CS operators. We also show that the 3-qubit Clifford+CS group, which is of course infinite, is the amalgamated product of three finite subgroups.
Quantum computations starting with computational basis states and involving only Clifford operations, are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup GGG of U(d)U(d)U(d). In particular we seek GGG such that G⊗GG\otimes GG⊗G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2d = 2d=2 we exhaustively treat all finite irreducible subgroups of U(2)U(2)U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups generated by X and the nthn^{\rm th}nth root of ZZZ for n∈Nn \in \Nn∈N.
As two basic building blocks for any quantum circuit, we consider the 1-qubit PHASOR circuit ϕ( θ ) and the 1-qubit NEGATOR circuit N ( θ ). Both are roots of the IDENTITY circuit. Indeed: both ϕ(0) and N (0) equal the 2 × 2 unit matrix. Additionally, the NEGATOR is a root of the classical NOT gate. Quantum circuits (acting on w qubits) consisting of controlled PHASORs are represented by matrices from ZU(2 w ); quantum circuits consisting of controlled NEGATORs are represented by matrices from XU(2 w ). Here, ZU( n ) and XU( n ) are subgroups of the unitary group U( n ): the group XU( n ) consists of all n × n unitary matrices with all 2 n line sums (i.e. all n row sums and all n column sums) equal to 1 and the group ZU( n ) consists of all n × n unitary diagonal matrices with first entry equal to 1. Any U( n ) matrix can be decomposed into four parts: U = exp( iα ) Z 1 XZ 2 , where both Z 1 and Z 2 are ZU( n ) matrices and X is an XU( n ) matrix. We give an algorithm to find the decomposition. For n = 2 w it leads to a four-block synthesis of an arbitrary quantum computer.
Quantum computations that involve only Clifford operations are classically simulable despite the fact that they generate highly entangled states; this is the content of the Gottesman-Knill theorem. Here we isolate the ingredients of the theorem and provide generalisations of some of them with the aim of identifying new classes of simulable quantum computations. In the usual construction, Clifford operations arise as projective normalisers of the first and second tensor powers of the Pauli group. We consider replacing the Pauli group by an arbitrary finite subgroup G of U(d). In particular we seek G such that G tensor G has an entangling normaliser. Via a generalisation of the Gottesman-Knill theorem the resulting normalisers lead to classes of quantum circuits that can be classically efficiently simulated. For the qubit case d=2 we exhaustively treat all finite subgroups of U(2) and find that the only ones (up to unitary equivalence and trivial phase extensions) with entangling normalisers are the groups G_n generated by X and the n^th root of Z.
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).