Quantum Expander Mixing Lemma and its Converse

Expander graphs are foundational in both computer science and mathematics, offering diverse applications across these domains (Hoory et al.,, 2006; Lubotzky,, 2012). Leveraging their expansion property, these graphs contribute significantly to algorithmic innovations, cryptographic protocols, analysis of circuit and proof complexity, development of derandomization techniques and pseudorandom generators, as well as the theory of error-correcting codes. Additionally, expander graphs play pivotal roles in shaping structural insights within group theory, algebra, number theory, geometry, and combinatorics. As quantum technology is revolutionizing the world, quantum expanders were introduced and have foundd numerous applications in the field of quantum information theory (Ambainis and Smith,, 2004; Ben-Aroya et al.,, 2008; Hastings, 2007a, ; Hastings and Harrow,, 2009; Aharonov et al.,, 2014).

Quantum expanders are the quantum extension of expander graphs, described by means of operators that map quantum states to quantum states. A general quantum state is a density matrix, which is a trace-one and positive semidefinite operator, i.e.,

with $\left\{\psi_{v}\right\}$ being some orthonormal basis of a Hilbert space $\mathcal{V}$. Notice that $\rho\in L(\mathcal{V}):=\operatorname{Hom}(\mathcal{V},\mathcal{V})$. An admissible quantum transformation $F:L(\mathcal{V})\rightarrow L(\mathcal{V})$ is any transformation that can be implemented by a quantum circuit, with unitary operators and measurements. Thus, admissible quantum operators map density matrices to density matrices; see, Nielsen and Chuang, (2001); Kitaev et al., (2002).

In this paper, we rigorously define quantum expanders following Hastings, 2007b and Pisier, (2014). In essence, we consider a quantum operator $T_{n}:M(N_{n})\rightarrow M(N_{n})$ in the following normalized form (dividing by $d$):

where $M(N_{n})$ stands for the space of $N_{n}\times N_{n}$ complex matrices, $\big{(}u_{1}^{(n)},\cdots,u_{d}^{(n)}\big{)}$ is a $d$-tuple of unitary matrices with $d$ being a positive integer that is independent of $n$, and the superscript $``\ast"$ represents the conjugate transpose. We have the operator norm $\|T_{n}\|=1$ and $T_{n}(I_{N_{n}})=I_{N_{n}}$, where $I_{N_{n}}\in M(N_{n})$ is the identity matrix. In the quantum context, the spectral gap is delineated by the reduced spectral radius $\big{\|}T_{n}|_{I_{N_{n}}^{\perp}}\big{\|}$, where $T_{n}$ is restricted to the orthogonal complement $I_{N_{n}}^{\perp}$ of the identity matrix. A quantum expander sequence comprises those having reduced spectral radius smaller than $1$ uniformly for all $n$, as $N_{n}$ tends to infinity with $n$.

The aim of this paper is to establish the quantum counterpart of the classical expander mixing lemma and its converse. The expander mixing lemma intuitively states that the edges of certain $d$-regular graphs are evenly distributed throughout the graph. Specifically, the normalized number of edges connecting two vertex subsets $S_{1}$ and $S_{2}$ is always close to the expected number of edges between them in a random $d$-regular graph, which is $\frac{1}{n}|S_{1}||S_{2}|$. This well-known result is demonstrated, for example, in Corollary 9.2.5 of Alon and Spencer, (2000). The expander mixing lemma plays a pivotal role in graph theory, providing crucial insights into the distribution of edges within graphs and facilitating the analysis of various network properties and algorithms. Variants of the expander mixing lemma have been proposed. For instance, Chen et al., (2013) established an operator version of the expander mixing lemma, utilizing it iteratively for bias amplification. This approach was further employed in Jeronimo et al., (2022) to formulate the iterated operator expander mixing lemma.

We establish the expander mixing lemma for quantum expanders, termed the quantum expander mixing lemma, for the first time to the authors’ best knowledge. This lemma asserts that for any two orthogonal projections $P_{1},P_{2}\in M(N_{n})$, the Hilbert–Schmidt inner product of $P_{1}$ and $T_{n}P_{2}$ is always close to $\frac{1}{N_{n}}{\operatorname{tr}}(P_{1}){\operatorname{tr}}(P_{2})$ for all $n$, for $T_{n}$ being in the normalized form given in (1.1). In quantum mechanics, the Hilbert-Schmidt inner product of two matrices (which symbolize quantum states) serves as a measure of the similarity between these states. A substantial inner product implies a significant degree of similarity or correlation between the states, whereas a small inner product indicates dissimilarity or orthogonality.

Studying the converse of the expander mixing lemma enhances the understanding of graph properties and their implications in diverse fields, leading to more robust theories, algorithms, and applications. The converse of the expander mixing lemma has been articulated in two distinct ways. The first way involves investigating the upper bound of the second-largest (in absolute value) eigenvalue of the $d$-regular graph, when the normalized number of edges between two vertex subsets $S_{1}$ and $S_{2}$ is always close to $\frac{1}{n}|S_{1}||S_{2}|$, as demonstrated in Bilu and Linial, (2006). The second way examines whether there exist vertex subsets $S_{1}$ and $S_{2}$ such that the normalized number of edges in between, compared to $\frac{1}{n}|S_{1}||S_{2}|$, cannot fall below a certain threshold, as shown in Lev, (2015).

In this paper, we further establish the converse of the expander mixing lemma for quantum expanders, which is the first result to the best of the authors’ knowledge. Following the second way, our focus lies on exploring the bounds of that difference. Specifically, our quantum expander mixing lemma states that there exist two orthogonal projections $P_{1},P_{2}\in M(N_{n})$ such that the difference of the Hilbert–Schmidt inner product of $P_{1}$ and $T_{n}P_{2}$ to $\frac{1}{N_{n}}{\operatorname{tr}}(P_{1}){\operatorname{tr}}(P_{2})$ cannot be small up to a certain threshold for all $n$.

The rest of the paper proceeds as follows. In Section 2, we formally define expander graphs and then present the classical expander mixing lemma. In Section 3, we rigorously define quantum expander graphs and then state our quantum expander mixing lemma along with its converse. Technical proofs of this paper are provided in Section 4. Throughout the paper, $\|\cdot\|$ denotes the Hilbert–Schmidt norm when applied to a matrix and the operator norm when applied to an operator, with the specific meaning determined by the context.

We start by defining formally the concept of an expander graph and then state the expander mixing lemma in Theorem 2.3. There exist several equivalent definitions of expander graphs, which can be characterized through vertex, edge, or spectral expansion. We adopt the spectral perspective towards expander graphs, defining them in terms of a certain spectral gap by means of Markov operators. We refer interested readers to Tao, (2015) for further details.

We consider an undirected graph $\mathcal{G}=(V,E)$, where $V$ is the set of vertices and $E$ is the set of edges. A graph is finite if $V$ is finite, and hence $E$ is finite. We consider $\mathcal{G}$ as a $d$-regular graph, where each vertex of $V$ is contained in exactly $d$ edges in $E$; we say that $d$ is the degree of the regular graph $\mathcal{G}$. We let $\ell^{2}(V)$ be the finite-dimensional complex Hilbert space of functions $f:V\rightarrow\mathbb{C}$ with norm and inner product given respectively as

where the overline notation represents the complex conjugate. We then define the adjacency operator $A:\ell^{2}(V)\rightarrow\ell^{2}(V)$ on functions $f\in\ell^{2}(V)$ as the sum of $f$ over all of the neighbours of $v$, i.e.,

Clearly, $A$ is a linear operator and one can associate it with the adjacency matrix.

Since $\mathcal{G}$ is $d$-regular, a linear algebraic definition of expansion is possible based on the eigenvalues of the adjacency matrix $A$. For $\mathcal{G}$ being undirected, $A$ is real symmetric. It is known that for $\mathcal{G}$ being a $d$-regular graph having $n$ vertices, the spectral theorem implies that $A$ has $n$ real-valued eigenvalues $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}$ and these eigenvalues are in $[-d,d]$. Because $\mathcal{G}$ is regular, the uniform distribution $u\in\mathbb{R}^{n}$, with entry $u_{i}=1/n$ for all $i=1,…,n$, is its stationary distribution. Thus, $u$ is an eigenvector of $A$ with eigenvalue $\lambda_{1}=d$, i.e., $Au=du$. The spectral gap of $\mathcal{G}$ is defined to be $d-\lambda_{2}$, which measures the spectral expansion.

The normalized variants of these definitions are commonly employed and offer greater convenience when displaying certain results. Consider the matrix $\frac{1}{d}A$, which is the Markov transition matrix of $\mathcal{G}$, and its eigenvalues are in $[-1,1]$. It is associated with the Markov operator $T:\ell^{2}(V)\rightarrow\ell^{2}(V)$ defined by

where $\delta_{x}$ is the Kronecker delta function on $x$. Then, $T$ is self-adjoint (equal to its conjugate transpose $T^{*}$) and the operator norm $\|T\|=1$. Denote $1_{V}\in\ell^{2}(V)$ for the constant function $v\mapsto 1$, and then $T(1_{V})=1_{V}$. We can observe the reduction in the operator norm of $T$ when it is restricted to the orthogonal complement $1_{V}^{\perp}$.

In this paper, we assume graph $\mathcal{G}$ is connected, and then the largest eigenvalue $1$ of $T$ has multiplicity $1$. We assume $\mathcal{G}$ is non-bipartite, and then $-1$ is not an eigenvalue of $T$. Therefore, the reduced spectral radius $\rho(\mathcal{G})$ is always smaller than $1$. We identify $\mathcal{G}$ with the set of vertices. A sequence $(\mathcal{G}_{n})_{n=1}^{\infty}$ is a sequence of finite $d$-regular simple connected graphs with the cardinality of the vertex set $|V_{n}|=n\rightarrow\infty$. For a given sequence of $d$-regular graphs $(\mathcal{G}_{n})_{n=1}^{\infty}$, the reduced spectral radius $\rho(\mathcal{G}_{n})$ for each $n\in\mathbb{N}$ is always smaller than $1$. The operator $\Delta:=1-T$ is sometimes known as the (combinatorial or graph) Laplacian. This is a positive semidefinite operator with at least one zero eigenvalue. To inquire whether a uniform gap exists between $\rho(\mathcal{G}_{n})$ and $1$, here is the definition of the expander sequence.

We have defined the notion of an expander sequence, which is a family of graphs instead of an individual graph. It is also commonly defined through the $(n,d,\lambda)$-graph, which is a $d$-regular graph with $n$ vertices such that all of the eigenvalues of its adjacency matrix except one have absolute value at most $\lambda$, i.e., $\max_{i\neq 1}|\lambda_{i}|\leq\lambda$. If we fix $d$ and $\lambda$ then $(n,d,\lambda)$-graphs form a family of expander graphs with a constant spectral gap. The expander mixing lemma states the number of edges between two vertex subsets $S_{1}$ and $S_{2}$ is always close to the expected number of edges between them in a random $d$-regular graph, namely $\frac{d}{n}|S_{1}||S_{2}|$; see, for instance, Corollary 9.2.5 of Alon and Spencer, (2000).

A converse to the above well-known expander mixing lemma is provided in Corollary 4 of Lev, (2015).

In this section, we define formally the concept of quantum expanders following Hastings, 2007b and Pisier, (2014). Before that, it is worth providing the motivation from classical expanders and explaining how quantum expanders appear as a non-commutative version of the classical ones.

The story could be started with the Cayley graph. Consider $G$ as a finite group generated by $S=\{s_{1},\ldots,s_{d}\}$. Suppose $S$ is symmetric in the sense that $s\in S$ whenever $s^{-1}\in S$ and does not contain the identity $1$ to avoid loops. The Cayley graph $\operatorname{Cay}(G,S)$ is defined as the graph with vertex set $G$ and edge set $\{\{sx,x\}:x\in G,s\in S\}$. Each vertex $x\in G$ is connected to the $|S|=d$ elements $sx$ for $s\in S$ and hence $\operatorname{Cay}(G,S)$ is a $d$-regular graph. A unitary representation of $G$ is the Hilbert space $\ell^{2}(G)$, together with a homomorphism $\rho:G\rightarrow U(\ell^{2}(G))$ from $G$ to the group $U(\ell^{2}(G))$ of unitary transformations on $\ell^{2}(G)$. Define the (left) regular unitary representation of $G$ as $\pi_{G}:G\rightarrow U(\ell^{2}(G))$ such that

for $f\in l^{2}(G)$ and $x,g\in G$. Unitary operator $\pi_{G}(s)$ is a unitary induced by the permutation of vertices of the graph.

To discuss expanders, let us assume that we are given a sequence of Cayley graphs ${\operatorname{Cay}(G_{n},S_{n})}_{n=1}^{\infty}$, where $S_{n}=\{s_{1}(n),\cdots s_{d}(n)\}\subset G_{n}$ for each $n$ with $d$ being a positive integer independent of $n$. Then $\{\operatorname{Cay}(G_{n},S_{n})\}_{n=1}^{\infty}$ where

is an expander sequence if and only if the sequence of $d$-tuples of unitaries

satisfies that with $\epsilon>0$,

where $\|\cdot\|$ is the Hilbert–Schmidt norm. This is equivalent to the condition that there exists a sequence of $d$-tuples of unitaries

such that for any $n\geq 1$, some $\epsilon>0$, and any $x$ being a $N_{n}\times N_{n}$ diagonal complex matrix,

where $U(N_{n})$ stands for the set of $N_{n}\times N_{n}$ unitary matrices.

The term “quantum expander” is just to designate unitaries $\big{(}u_{1}^{(n)},\cdots,u_{d}^{(n)}\big{)}$ to satisfy (3.1) for a general $x\in M(N_{n})$ beyond being diagonal, where $M(N_{n})$ stands for the space of $N_{n}\times N_{n}$ complex matrices. In this light, quantum expanders can be seen as a non-commutative version of the classical ones. Identifying $M(N_{n})$ with the space $B(\ell_{2}^{N_{n}})$ of bounded operators on the $N_{n}$-dimensional Hilbert space $\ell_{2}^{N_{n}}$, different to the Markov operator $T$ defined in (2.3), in the quantum setting we consider quantum operators act on $\ell_{2}^{N_{n}}\otimes\overline{\ell_{2}^{N_{n}}}$ of the following form for each $n$:

where $\overline{u}_{j}^{(n)}$ is the complex conjugate of the matrix $u_{j}^{(n)}$. Identifying as usual $\ell_{2}^{N_{n}}\otimes\overline{\ell_{2}^{N_{n}}}$ as $S_{2}^{N_{n}}$, the Hilbert space obtained by equipping $M(N_{n})$ with the corresponding scalar product and the Hilbert–Schmidt norm. We can write

Then we may consider $T_{n}$ as an operator acting on $M(N_{n})$ defined by

which satisfies that

Analogous to the classical setting, we have the operator norm $\|T_{n}\|=1$ and $T_{n}(I_{N_{n}})=I_{N_{n}}$, where $I_{N_{n}}\in M(N_{n})$ is the identity matrix. Then we ask how much the operator norm of the quantum operator $T_{n}$ would be decreased if it is restricted to the orthogonal complement $I_{N_{n}}^{\perp}$ of the identity matrix. For that purpose, we give the quantum version of Definition 2.1.

The definition of a quantum expander is defined in terms of $\rho(n)$. It says that there exists $\epsilon>0$ such that $\rho(n)\leq 1-\epsilon$ for all $n$, which is in consistent with the classical expander sequence in Definition 2.2.

The following two theorems are our main results of this paper.

To prove Theorem 3.4, we commence by introducing the Schatten norm. In mathematics, particularly in functional analysis, the Schatten norm, also known as the Schatten–von-Neumann norm, emerges as a generalization of p-integrability, akin to the trace class norm and the Hilbert–Schmidt norm. For $p\in[1,\infty)$, define the Schatten $p$-norm of $A\in M(N_{n})$ as

for $p=\infty$, the convention is to define $\|\cdot\|_{\infty}$ as the operator norm

The Schatten norms are unitarily invariant in the sense that for unitaries $U,V$ and $p\in[1,\infty]$,

Considering $q=1$ as the dual index to $p=\infty$, the noncommutative Hölder’s inequality says that for all $p,q,r\in[1,\infty]$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, we have that for any matrices $A,B\in M(N_{n})$,

Its special case is the well-known inequality

where $\|\cdot\|_{1}$ is called the trace norm or the nuclear norm, $\|\cdot\|_{2}$ is called the Frobenius norm or the Hilbert–Schmidt norm, and $\|\cdot\|_{\infty}$ is called the spectral norm or the operator norm. Lev, (2015) defined height (Definition 4.1) of a matrix using these three important norms; by (4.2) the height of any matrix is at least one.

We define the Schatten norm induced operator norm as $\|T_{n}\|_{p\rightarrow p}$ for operator $T_{n}:M(N_{n})\rightarrow M(N_{n})$. Then the three special cases $p=1,2$ and $\infty$ give operator norms $\|T\|_{1\rightarrow 1}$, $\|T\|_{2\rightarrow 2}$ and $\|T\|_{\infty\rightarrow\infty}$, respectively. Then we define a linear map for unitaries $U_{1},U_{2},V_{1},V_{2}\in M(N_{n})$ as follows:

where $D_{\lambda}$ is the diagonal matrix ${\operatorname{diag}}(\lambda_{1},\cdots\lambda_{n})$. Here, we follow the convention that when the ${\operatorname{diag}}$ operator is applied to a vector, it yields the corresponding diagonal matrix, and when applied to a matrix, it produces the corresponding diagonal vector. Then we have

The next lemma is a special case of Gillespie, (1991) on page 226 therein.

Similar to the definition of height for a matrix in Definition 4.1, we give definition of the height of a linear operator here; by Lemma 4.2 the height of any linear operator is at least one.

Recall that in the classical setting, for an operator $T:\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$, we have

In the quantum setting, denote $T_{n}^{*}$ as the adjoint of $T_{n}:M(N_{n})\rightarrow M(N_{n})$ with respect to the Hilbert–Schmidt inner product. For $\lambda,\mu\in\mathbb{C}^{n}$, we have

which yields that

Hence, we have

By Definition 4.3,

In the classical setting, Lev, (2015) defined the logarithmic diameter for a non-zero vector $z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ by

In the quantum setting, considering complex matrices, we define the logarithmic diameter as follows.

The following lemma will serve as the cornerstone for the subsequent proofs.