Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

TL;DR

This paper characterizes certain restricted Clifford+T quantum circuits using number theory, linking matrix entries over specific rings to well-known universal gate sets, advancing quantum circuit synthesis understanding.

Contribution

It extends number-theoretic characterizations to subrings of [1/2, i], connecting matrix entries to specific universal gate sets.

Findings

Unitary matrices over [1/2] correspond to circuits with classical gates plus Hadamard and phase gates.

Matrices over [1/2] relate to circuits with classical gates, Hadamard, and phase gates.

Results provide new insights into the structure of restricted Clifford+T circuits.

Abstract

Kliuchnikov, Maslov, and Mosca proved in 2012 that a $2\times 2$ unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\mathbb{Z}[1/\sqrt{2},i]$. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+$T$ circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+$T$ circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+$T$ circuits by considering unitary matrices over subrings of $\mathbb{Z}[1/\sqrt{2},i]$. We focus on the subrings $\mathbb{Z}[1/2]$, $\mathbb{Z}[1/\sqrt{2}]$, $\mathbb{Z}[1/i\sqrt{2}]$, and $\mathbb{Z}[1/2,i]$, and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates $\{X, CX, CCX\}$ with an analogue of the Hadamard gate and an optional phase gate.

Equations

$$H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right],CX=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right],\text{mbox}andT=\left[\begin{array}{cc}1& 0\\ 0& \omega \end{array}\right].$$

$$G_{1}\cdots G_{\ell}\left[\begin{array}[]{c|ccc}&&&\\ v&&V^{\prime}&\\ &&&\end{array}\right]=\left[\begin{array}[]{c|ccc}1&0&\cdots&0\\ \hline\cr 0&&&\\ \vdots&&V^{\prime\prime}&\\ 0&&&\end{array}\right]$$

$$V=\left[\begin{array}{cc}{v}_{1,1}& v_{1,2}\\ v_{2,1}& v_{2,2}\end{array}\right]$$

$$V_{[1,3]}=\left[\begin{array}{ccc}{v}_{1,1}& 0& v_{1,2}\\ 0& 1& 0\\ v_{2,1}& 0& v_{2,2}\end{array}\right]$$

$${W_{[a_1,\dots ,a_n]}}_{j,k}=\{\begin{array}{l}{W}_{j^{\prime },k^{\prime }}\text{mbox}ifj=a_{j^{\prime }}\text{mbox}andk=a_{k^{\prime }}\\ I_{j,k}\text{mbox}otherwise.\end{array}$$

$$V=\frac{1}{p^q}W$$

$$p^{q^{\prime }}V\in R^{m\times n}.$$

$$P_{\zeta }=\left[\begin{array}{cc}1& 0\\ 0& \zeta \end{array}\right].$$

$$Z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],S=\left[\begin{array}{cc}1& 0\\ 0& i\end{array}\right],\text{mbox}andT=\left[\begin{array}{cc}1& 0\\ 0& \omega \end{array}\right].$$

$$H=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& 1\\ 1& -1\end{array}\right]\text{mbox}andX=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right].$$

$$F=\frac{1}{2}\left[\begin{array}{cc}1+i\sqrt{2}& 1\\ 1& -1+i\sqrt{2}\end{array}\right].$$

$$F=SHTSHTSHS{\omega }^{-1}.$$

$$CH=I_2\oplus H,CX=I_2\oplus X,\text{mbox}andCCX=I_6\oplus X.$$

$$D|\psi ⟩{|0⟩}^{\otimes m}=(W|\psi ⟩){|0⟩}^{\otimes m}.$$

$$D|\psi ⟩|\varphi ⟩=(W|\psi ⟩)|\varphi ⟩.$$

Full text

1 Introduction

Kliuchnikov, Maslov, and Mosca showed in [KMM-exact] that a 2-dimensional unitary matrix $V$ can be exactly represented by a single-qubit Clifford+$T$ circuit if and only if the entries of $V$ belong to the ring $\mathbb{Z}[1/\sqrt{2},i]$. This result gives a number-theoretic characterization of single-qubit Clifford+$T$ circuits. In [GS13], Giles and Selinger extended the characterization of Kliuchnikov et al. to multi-qubit Clifford+$T$ circuits by proving that a $2^{n}$-dimensional unitary matrix can be exactly represented by an $n$-qubit Clifford+$T$ circuit if and only if its entries belong to $\mathbb{Z}[1/\sqrt{2},i]$.

These number-theoretic characterizations provide great insight into the structure of Clifford+$T$ circuits. As a result, single-qubit Clifford+$T$ circuits are now very well understood [fallback, ma-remarks, kmm-approx, MA08, RS16], and some of these results have even been extended to single-qubit circuits beyond the Clifford+$T$ gate set [bgs13, fgkm15, kbs14, ky15, ps18, r15]. In contrast, our understanding of multi-qubit Clifford+$T$ circuits remains more limited, despite interesting results [BS2015, devos, GKMR, Gr2014, BSW2016]. One of the reasons for this limitation is that large unitary matrices over $\mathbb{Z}[1/\sqrt{2},i]$ are hard to analyze. In order to circumvent the difficulties associated with multi-qubit Clifford+$T$ circuits, restricted gate sets have been considered in the literature. This led to important developments in the study of multi-qubit Clifford, CNOT+$T$, and CNOT-dihedral circuits [ACR17, AMM, AMMR, AM, CH2018, MSdM2018, Sel]. Unfortunately, the simpler structure of these restricted gate sets comes at a cost: they are not universal for quantum computing.

In the present paper, our goal is to address both of these limitations by considering restrictions of the Clifford+$T$ gate set which are nevertheless universal for quantum computing. To this end, we study circuits corresponding to unitary matrices over proper subrings of $\mathbb{Z}[1/\sqrt{2},i]$, focusing on $\mathbb{Z}[1/2]$, $\mathbb{Z}[1/\sqrt{2}]$, $\mathbb{Z}[1/i\sqrt{2}]$, and $\mathbb{Z}[1/2,i]$. For each subring, we find a set of quantum gates $G$ with the property that circuits over $G$ correspond to unitary matrices over the given ring. Writing $U_{2^{n}}(R)$ for the group of $2^{n}\times 2^{n}$ unitary matrices over a ring $R$, our main results can then be summarized in the following theorem.

Theorem**.**

A $2^{n}\times 2^{n}$ unitary matrix $V$ can be exactly represented by an $n$-qubit circuit over

1. (i)

$\left\{X,CX,CCX,H\otimes H\right\}$* if and only if $V\in U_{2^{n}}(\mathbb{Z}[1/2])$,* 2. (ii)

$\left\{X,CX,CCX,H,CH\right\}$* if and only if $V\in U_{2^{n}}(\mathbb{Z}[1/\sqrt{2}])$,* 3. (iii)

$\left\{X,CX,CCX,F\right\}$* if and only if $V\in U_{2^{n}}(\mathbb{Z}[1/i\sqrt{2}])$, and* 4. (iv)

$\left\{X,CX,CCX,\omega H,S\right\}$* if and only if $V\in U_{2^{n}}(\mathbb{Z}[1/2,i])$,*

where $\omega=e^{i\pi/4}$ and $F\propto\sqrt{H}$. Moreover, in (i)-(iv), a single ancilla is sufficient.

The gate sets in items $(i)$–$(iv)$ of the above theorem are all universal for quantum computing [Aharonov03asimple, Shi2003], and we sometimes refer to circuits over these gate sets as integral, real, imaginary, and Gaussian Clifford+$T$ circuits, respectively. As a corollary to the above theorem, we also obtain two additional characterizations of universal gate sets.

Corollary**.**

A $2^{n}\times 2^{n}$ unitary matrix $V$ can be exactly represented by an $n$-qubit circuit over

1. (i)

$\left\{X,CX,CCX,H\right\}$* if and only if $V=W/\sqrt{2}^{q}$ for some matrix $W$ over $\mathbb{Z}$ and some $q\in\mathbb{N}$, and* 2. (ii)

$\left\{X,CX,CCX,H,S\right\}$* if and only if $V=W/\sqrt{2}^{q}$ for some matrix $W$ over $\mathbb{Z}\left[i\right]$ and some $q\in\mathbb{N}$.*

Moreover, in (i) and (ii), a single ancilla is sufficient.

As a final corollary to the theorem above, we refine the characterizations $(iii)$ and $(iv)$ by showing that in these cases a matrix can be represented by an ancilla-free circuit if and only if it has determinant 1.

Corollary**.**

Let $n\geq 4$. A $2^{n}\times 2^{n}$ unitary matrix $V$ can be exactly represented by an $n$-qubit ancilla-free circuit over

1. (i)

$\left\{X,CX,CCX,F\right\}$* if and only if $V\in U_{2^{n}}(\mathbb{Z}[1/i\sqrt{2}])$ and $\det V=1$, and* 2. (ii)

$\left\{X,CX,CCX,\omega H,S\right\}$* if and only if $V\in U_{2^{n}}(\mathbb{Z}[1/2,i])$ and $\det V=1$.*

In (i) and (ii), the requirement that $\det V=1$ can be dropped for $n<4$.

The characterization of ancilla-free real and integral Clifford+$T$ circuits remains an open question but we conjecture that they correspond to a strict subgroup of the group of unitaries with determinant 1.

Restrictions similar to the ones considered here were previously studied in the context of foundations [RL2002], randomized benchmarking [HFGW18], and graphical languages for quantum computing [BK2019, JPV2017, vilmart2019]. Furthermore, our study fits within a larger program, initiated by Aaronson and others, which aims at classifying quantum operations. Such classifications exist for classical reversible operations [AGS17], for stabilizer operations [GS18], and for beam-splitter interactions [ab], but no classification is known for a universal family of quantum operations suited for fault-tolerant quantum computing. In this context, our work can be seen as a partial classification of the universal extensions of the set of classical reversible gates $\left\{X,CX,CCX\right\}$. This perspective is illustrated in Figure 1, which depicts a fragment of the lattice of subgroups of $U_{n}(\mathbb{Z}[1/\sqrt{2},i])$ where, for conciseness, we wrote $\mathbb{D}$ for the ring $\mathbb{Z}[1/2]$ so that the rings $\mathbb{Z}[1/\sqrt{2}]$, $\mathbb{Z}[1/i\sqrt{2}]$, $\mathbb{Z}[1/2,i]$ and $\mathbb{Z}[1/\sqrt{2},i]$ are denoted by $\mathbb{D}\left[\sqrt{2}\right]$, $\mathbb{D}\left[i\sqrt{2}\right]$, $\mathbb{D}[i]$, and $\mathbb{D}\left[\omega\right]$, respectively.

The rest of the paper is organized as follows. In Section 2, we give an overview of our methods. In Section 3, we introduce the rings and matrices which will be used throughout the paper. In Section 4, we show that certain useful matrices can be exactly represented by restricted Clifford+$T$ circuits. LABEL:sec:characterizations contains the proofs of our various number-theoretic characterizations. We conclude in LABEL:sec:conc.

2 Overview

Unrestricted Clifford+$T$ circuits are generated by

[TABLE]

Since $\omega=(1+i)/\sqrt{2}$, the entries of all the generators belong to the ring $\mathbb{Z}[1/\sqrt{2},\omega]=\mathbb{Z}[1/\sqrt{2},i]=\mathbb{D}[\omega]$. Hence, if a matrix $V$ can be represented exactly by an $n$-qubit Clifford+$T$ circuit, then $V\in U_{2^{n}}(\mathbb{D}\left[\omega\right])$, the group of $2^{n}\times 2^{n}$ unitary matrices with entries in $\mathbb{D}\left[\omega\right]$. Showing that the ring $\mathbb{D}\left[\omega\right]$ characterizes Clifford+$T$ circuits thus amounts to proving the converse implication. An algorithm establishing that every element of $U_{2^{n}}(\mathbb{D}\left[\omega\right])$ can be exactly represented by a Clifford+$T$ circuit is known as an exact synthesis algorithm.

The original insight of Kliuchnikov, Maslov and Mosca in the single-qubit Clifford+$T$ case was to reduce the problem of exact synthesis to the problem of state preparation. The latter problem is to find, given a target vector $v\in\mathbb{D}\left[\omega\right]^{n}$, a sequence $G_{1},\ldots,G_{\ell}$ of Clifford+$T$ gates such that $G_{\ell}\cdots G_{1}e_{1}=v$ or, equivalently, such that $G_{1}^{\dagger}\cdots G_{\ell}^{\dagger}v=e_{1}$. Kliuchnikov et al. realized that this sequence of gates can be found by first writing $v$ as $v=u/\sqrt{2}^{q}$ for some $u\in\mathbb{Z}\left[\omega\right]$ and then iteratively reducing the exponent $q$.

This basic premise was extended by Giles and Selinger to the multi-qubit context by adding an outer induction over the columns of an $n$-qubit unitary. This method amounts to performing a constrained Gaussian elimination where the row operations are restricted to a few basic moves. The Giles-Selinger algorithm proceeds by reducing the leftmost column of an $n\times n$ unitary matrix to the first standard basis vector by applying a sequence of one- and two-level matrices, which act non-trivially on at most two components of a vector, before recursively dealing with the remaining submatrix. If the target unitary is V=\left[\begin{array}[]{c|c}v&V^{\prime}\end{array}\right], then the Giles-Selinger algorithm first constructs a sequence of matrices $G_{1},\dots,G_{\ell}$ such that $G_{1}\cdots G_{\ell}v=e_{1}$. Left-multiplying $V$ by this sequence of matrices then yields

[TABLE]

where $V^{\prime\prime}$ is unitary. The fact that the matrices used in this reduction act non-trivially on no more than two rows of the matrix ensures that when the algorithm recursively reduces the columns of $V^{\prime\prime}$ it does so without perturbing the previously fixed columns. The Giles-Selinger algorithm thus relies on the following two facts.

A unit vector in $\mathbb{D}\left[\omega\right]^{n}$ can be reduced to a standard basis vector by using one- and two-level matrices and 2. 2.

the required one- and two-level matrices can be exactly represented by Clifford+$T$ circuits.

While there are subtle differences between the various cases discussed below, our method in characterizing restricted Clifford+$T$ circuits follows this general structure.

3 Rings and Matrices

In this section, we discuss the rings and matrices that will be used throughout the paper. For further details, the reader is encouraged to consult [artin].

3.1 Rings

We write $\mathbb{N}$ for the set of nonnegative integers and if $n\in\mathbb{N}$ we write $[n]$ for the set $\left\{1,\ldots,n\right\}$. We use $\mathbb{Z}$ to denote the ring of integers and $i$ to denote the imaginary unit. We define $\omega$ as $\omega=e^{i\pi/4}=(1+i)/\sqrt{2}$. Note that $i$ is a 4-th root of unity and that $\omega$ is an 8-th root of unity.

We will use the extensions of $\mathbb{Z}$ defined below.

Definition 3.1**.**

Let

• •

$\mathbb{Z}\left[\sqrt{2}\right]=\left\{x_{0}+x_{1}\sqrt{2}\mid x_{0},x_{1}\in\mathbb{Z}\right\}$,

• •

$\mathbb{Z}\left[i\sqrt{2}\right]=\left\{x_{0}+x_{1}i\sqrt{2}\mid x_{0},x_{1}\in\mathbb{Z}\right\}$,

• •

$\mathbb{Z}\left[i\right]=\left\{x_{0}+x_{1}i\mid x_{0},x_{1}\in\mathbb{Z}\right\}$, and

• •

$\mathbb{Z}\left[\omega\right]=\left\{x_{0}+x_{1}\omega+x_{2}\omega^{2}+x_{3}\omega^{3}\mid x_{0},x_{1},x_{2},x_{3}\in\mathbb{Z}\right\}$.

The rings $\mathbb{Z}\left[\sqrt{2}\right]$, $\mathbb{Z}\left[i\sqrt{2}\right]$, $\mathbb{Z}\left[i\right]$, and $\mathbb{Z}\left[\omega\right]$ are known as the ring of quadratic integers with radicand 2, the ring of quadratic integers with radicand -2, the ring of Gaussian integers, and the ring of cyclotomic integers of degree 8, respectively. All of these rings are distinct subrings of $\mathbb{Z}\left[\omega\right]$ and we have the inclusions depicted in the lattice of subrings below.

\mathbb{Z}$$\mathbb{Z}\left[i\sqrt{2}\right]$$\mathbb{Z}\left[i\right]$$\mathbb{Z}\left[\sqrt{2}\right]$$\mathbb{Z}\left[\omega\right]

Further to the rings introduced in Definition 3.1, we will consider extensions of the ring of dyadic fractions, i.e., fractions whose denominator is a power of 2.

Definition 3.2**.**

The ring of dyadic fractions $\mathbb{D}$ is defined as $\mathbb{D}=\left\{\frac{u}{2^{q}}\mid u\in\mathbb{Z},q\in\mathbb{N}\right\}$.

Definition 3.3**.**

Let

• •

$\mathbb{D}\left[\sqrt{2}\right]=\left\{x_{0}+x_{1}\sqrt{2}\mid x_{0},x_{1}\in\mathbb{D}\right\}$,

• •

$\mathbb{D}\left[i\sqrt{2}\right]=\left\{x_{0}+x_{1}i\sqrt{2}\mid x_{0},x_{1}\in\mathbb{D}\right\}$,

• •

$\mathbb{D}[i]=\left\{x_{0}+x_{1}i\mid x_{0},x_{1}\in\mathbb{D}\right\}$, and

• •

$\mathbb{D}\left[\omega\right]=\left\{x_{0}+x_{1}\omega+x_{2}\omega^{2}+x_{3}\omega^{3}\mid x_{0},x_{1},x_{2},x_{3}\in\mathbb{D}\right\}$.

If $v\in\mathbb{D}\left[\sqrt{2}\right]$, then $v$ can be written as $v=u/2^{q}$ for some $q\in\mathbb{N}$ and some $u\in\mathbb{Z}\left[\sqrt{2}\right]$. A similar property holds for elements of $\mathbb{D}\left[i\sqrt{2}\right]$, $\mathbb{D}[i]$, and $\mathbb{D}\left[\omega\right]$.

If $R$ is a ring and $r\in R$ we write $R/(r)$ for the quotient of the ring $R$ by the ideal generated by the element $r$. Two elements $s$ and $s^{\prime}$ of $R$ are congruent modulo $r$ if $s-s^{\prime}$ belongs to the ideal $(r)$, in which case we write $s\equiv s^{\prime}\pmod{r}$. We sometimes refer to the elements of the ring $R/(r)$ as residues. Some quotient rings are well-known. For example, $\mathbb{Z}/(2)=\left\{0,1\right\}$ and $\mathbb{Z}/(4)=\left\{0,1,2,3\right\}$. The following proposition gives an explicit description of certain lesser-known rings of residues which will be useful in what follows.

Proposition 3.4**.**

We have

• •

$\mathbb{Z}\left[\sqrt{2}\right]/(2)=\left\{0,1,\sqrt{2},1+\sqrt{2}\right\}$,

• •

$\mathbb{Z}\left[i\sqrt{2}\right]/(2)=\left\{0,1,i\sqrt{2},1+i\sqrt{2}\right\}$,

• •

$\mathbb{Z}\left[i\sqrt{2}\right]/(2i\sqrt{2})=\left\{0,1,2,3,i\sqrt{2},1+i\sqrt{2},2+i\sqrt{2},3+i\sqrt{2}\right\}$, and

• •

$\mathbb{Z}\left[i\right]/(2)=\left\{0,1,i,1+i\right\}$.

Proof.

To see, for example, that $\mathbb{Z}\left[\sqrt{2}\right]/(2)=\left\{0,1,\sqrt{2},1+\sqrt{2}\right\}$, note that $u=x_{0}+x_{1}\sqrt{2}$ and $u^{\prime}=x_{0}^{\prime}+x_{1}^{\prime}\sqrt{2}$ are congruent modulo 2 if there exists an element $t=y_{0}+y_{1}\sqrt{2}$ such that $u-u^{\prime}=2t$. This is the case if and only if $(x_{0}-x_{0}^{\prime})+(x_{1}-x_{1}^{\prime})\sqrt{2}=2y_{0}+2y_{1}\sqrt{2}$ which in turn holds if and only if $x_{0}\equiv x_{0}^{\prime}\pmod{2}$ and $x_{1}\equiv x_{1}^{\prime}\pmod{2}$. ∎

We will often take advantage of properties of residues. Some of these properties are generic. For example, if $u$ and $v$ are two elements of a ring $R$ and $u\equiv v\pmod{2}$, then $u\pm v\equiv 0\pmod{2}$. Other properties of residues are specific to a particular ring. For example, an integer $u\in\mathbb{Z}$ is odd if and only if $u^{2}\equiv 1\pmod{4}$. Similarly, for an integer $u\in\mathbb{Z}$, we have $u\equiv 3\pmod{4}$ if and only if $-u\equiv 1\pmod{4}$. We now state important properties of residues in $\mathbb{Z}\left[i\sqrt{2}\right]$ and $\mathbb{Z}\left[i\right]$ for future reference. They can be established by reasoning using residue tables in the relevant quotient rings. In the following, we denote the complex conjugate of an element $u$ by $u^{\dagger}$. For uniformity, we sometimes write $u^{\dagger}$ even when $u$ belongs to a real subring of $\mathbb{D}\left[\omega\right]$. In this case, $u^{\dagger}=u$.

Proposition 3.5**.**

The following statements hold.

• •

In $\mathbb{Z}\left[i\sqrt{2}\right]/(2)$, $u^{\dagger}u\equiv 0\mbox{ or }1$.

• •

If $u^{\dagger}u\equiv 1$ in $\mathbb{Z}\left[i\sqrt{2}\right]/(2)$, then $u\equiv 1,3,1+i\sqrt{2},\mbox{ or }3+i\sqrt{2}$ in $\mathbb{Z}\left[i\sqrt{2}\right]/(2i\sqrt{2})$.

• •

In $\mathbb{Z}\left[i\sqrt{2}\right]/(2i\sqrt{2})$, $u\equiv 3$ if and only if $-u\equiv 1$ and $u\equiv 3+i\sqrt{2}$ if and only if $-u\equiv 1+i\sqrt{2}$.

Proposition 3.6**.**

The following statements hold.

• •

In $\mathbb{Z}\left[i\right]/(2)$, if $u^{2}\equiv 1$, then $u\equiv 1\mbox{ or }i$.

• •

In $\mathbb{Z}\left[i\right]/(2)$, $u\equiv i$ if and only if $iu\equiv 1$.

3.2 Matrices

We write $e_{j}$ for the $j$-th standard basis vector and $M^{\dagger}$ for the conjugate transpose of the matrix $M$. If $R$ is a ring, we sometimes write $R^{n\times n^{\prime}}$ for the collection of $n\times n^{\prime}$ matrices over $R$. We will use one-, two-, and four-level matrices which act non-trivially on only one, two, or four of the components of their input. These matrices will be defined using basic matrices. The construction is best explained with an example. If

[TABLE]

is a 2-dimensional unitary matrix, then in 3 dimensions the two-level operator of type $V$, denoted by $V_{[1,3]}$, is the matrix given below.

[TABLE]

Definition 3.7**.**

Let $W$ be an $n\times n$ unitary matrix, let $n\leq n^{\prime}$, and let $a_{1},\ldots,a_{n}\in[n^{\prime}]$. The $n$-level matrix of type $W$ is the $n^{\prime}\times n^{\prime}$ unitary matrix $W_{[a_{1},\ldots,a_{n}]}$ defined by

[TABLE]

Let $R$ be one of $\mathbb{Z}$, $\mathbb{Z}\left[\sqrt{2}\right]$, $\mathbb{Z}\left[i\sqrt{2}\right]$, $\mathbb{Z}\left[i\right]$ or $\mathbb{Z}\left[\omega\right]$ and let $p$ be an element of $\mathbb{Z}\left[\omega\right]$. We will be interested in matrices of the form

[TABLE]

where $W$ is a matrix over $R$ and $q\in\mathbb{N}$.

Definition 3.8**.**

Fix $R\in\left\{\mathbb{Z},\mathbb{Z}\left[\sqrt{2}\right],\mathbb{Z}\left[i\sqrt{2}\right],\mathbb{Z}\left[i\right],\mathbb{Z}\left[\omega\right]\right\}$. If $V$ is a matrix of the form (1) and $q^{\prime}\in\mathbb{N}$, then we say that $q^{\prime}$ is a $p$-denominator exponent of $V$ if

[TABLE]

The smallest such $q^{\prime}$ is the least $p$-denominator exponent of $V$, denoted $\operatorname{lde}_{p}(V)$.

We sometimes omit $p$ when the base of the exponent is clear from the context. Note that the notion of denominator exponent applies to matrices of any dimension and we can therefore talk about the denominator exponent of a vector or scalar.

4 Circuits

In this section, we review basic circuit constructions which will be useful below. A more detailed discussion of quantum circuits can be found in Chapter 4 of [NC].

Let $\zeta$ be an $m$-th root of unity. We sometimes call $\zeta$ a global phase of order $m$. We think of these global phases as gates acting on 0 qubits and in what follows we will be especially interested in the global phases of order 2, 4, and 8, which we denote $-1$, $i$, and $\omega$, respectively. The single-qubit phase gate of order $m$ is defined as

[TABLE]

We will be particularly interested in phase gates of order 2, 4, and 8 which we call the $Z$, $S$, and $T$ gates, respectively. Hence

[TABLE]

In addition to phase gates, we will also use the single-qubit gates $H$ and $X$ defined by

[TABLE]

The $H$ gate is the Hadamard gate and the $X$ gate is the NOT gate. The last single-qubit gate we will use is the $F$ gate defined below.

[TABLE]

The $F$ gate is not as common as the other single-qubit gates introduced above. We note that $F^{2}=iH$ and that $F$ can be expressed as a product of better-known gates in Matsumoto-Amano normal form [MA08],

[TABLE]

We will also make use of the two-qubit $H\otimes H$ gate as well as the controlled gates defined below.

[TABLE]

We will refer to these gates as the controlled-$H$ gate, controlled-$X$ or CNOT gate, and the doubly-controlled-$X$ or Toffoli gate, respectively. In general, if $G$ is a gate, then we write $C^{n}G$ for the $n$-control-$G$ gate.

As usual, circuits are built from gates through composition and tensor product. An ancilla is a qubit used locally within a circuit but on which the global action of the circuit is trivial. We say that a $2^{n}\times 2^{n}$ unitary matrix $W$ is exactly represented by a circuit $D$ using $m$ clean ancillas if for any $n$-qubit state $\ket{\psi}$,

[TABLE]

If the circuit is independent of the initial state of the ancilla, it is said to accept a dirty ancilla. In particular, we say that a $2^{n}\times 2^{n}$ unitary matrix $W$ is exactly represented by a circuit $D$ using $m$ dirty ancillas if for any $n$-qubit state $\ket{\psi}$ and any $m$-qubit state $\ket{\phi}$,

[TABLE]

Note that a clean ancilla can always be used in place of a dirty ancilla.

In order to characterize restricted Clifford+$T$ circuits, it is helpful to establish some basic facts about the construction of multi-level matrices over gate sets including the Toffoli gate.

Proposition 4.1**.**

Any $2^{n}\times 2^{n}$ permutation matrix $V$ can be exactly represented by a circuit over the gate set $\{X,CX,CCX\}$ with at most one dirty ancilla.

Proof.

By the Giles-Selinger algorithm [GS13] restricted to permutation matrices, $V$ can be represented by a circuit over two-level $X$ gates. Likewise, each two-level $X$ gate can be implemented over fully-controlled $X$ and single-qubit gates by using a Gray code (see, e.g., [NC, Sec. 4.5.2]). Each fully-controlled $X$ gate can be implemented with one dirty ancilla by [BBC95], which completes the proof. ∎

Proposition 4.2**.**

Let $W$ be a $2^{m}\times 2^{m}$ unitary matrix and let $\mathcal{G}$ be a set of gates. If $CW$ can be exactly represented over $\{X,CX,CCX\}\cup\mathcal{G}$ using at most one dirty ancilla, then, for any $n\geq 1$, $C^{n}W$ can also be exactly represented over $\{X,CX,CCX\}\cup\mathcal{G}$. Moreover, a single clean ancilla suffices.

Proof.

This follows from standard techniques, e.g. [BBC95]. If $n=1$, then $CW$ can be implemented with a single dirty ancilla and thus also with a clean one. If $n>1$, then the $C^{n}W$ gate can be implemented as follows, where each gate on the right has at least one dirty ancilla available for use.

[TABLE]