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IEEE Conference on Computational Complexity (CCC)

Quantum Expanders: Motivation and Constructions

Avraham Ben-Aroya, Oded Schwartz, Amnon Ta‐Shma

Published Jun 1, 2008• 11 citations

Abstract

We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2, q) given by Lubotzky et al. (1988). The second construction is combinatorial, and is based on a quantum variant of the Zig-Zag product introduced by Reingold et al. (2000). Both constructions are of constant degree, and the second one is explicit. Using quantum expanders, we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK-complete (where QSZK is the class of languages having a zero-knowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof resembles the classical proof that the entropy difference problem is SZK-complete, but crucially depends on the use of quantum expanders.

DOI: 10.1109/ccc.2008.23

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We define quantum expanders in a natural way. We give two constructions of quantum expanders, both based on classical expander constructions. The first construction is algebraic, and is based on the construction of Cayley Ramanujan graphs over the group PGL(2, q) given by Lubotzky et al. (1988). The second construction is combinatorial, and is based on a quantum variant of the Zig-Zag product introduced by Reingold et al. (2000). Both constructions are of constant degree, and the second one is explicit. Using quantum expanders, we characterize the complexity of comparing and estimating quantum entropies. Specifically, we consider the following task: given two mixed states, each given by a quantum circuit generating it, decide which mixed state has more entropy. We show that this problem is QSZK-complete (where QSZK is the class of languages having a zero-knowledge quantum interactive protocol). This problem is very well motivated from a physical point of view. Our proof resembles the classical proof that the entropy difference problem is SZK-complete, but crucially depends on the use of quantum expanders.

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class NP, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class QSZK, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.

For every odd prime power

q

and

D=q+1

, we construct an infinite family of Ramanujan quantum expanders of degree

D

. The construction transfers Morgenstern's Ramanujan Cayley graphs on

\operatorname{PSL}_2(\mathbb{F}_{p})

for

p

which is an even power of

q

, through the odd irreducible subrepresentation of the Weil representation of

\operatorname{SL}_2(\mathbb{F}_{p})

. For a quantum expander of dimension

N

, our implementation uses

O(\log^2 N)

elementary gates, and

O(\log N)

ancilla qudits, using a fixed finite gate set depending on

q

. An advantage compared to the previous work of Iyer, Jain, Jordan, and Somma (FOCS 2026) is that assuming the quantum circuit is implemented exactly, we satisfy the

2\sqrt{D-1}/D

singular value bound exactly without any additive error.

We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders. Background. Classical expanders can be defined in either combinatorial or spectral terms, while quantum expanders usually have only a spectral definition. Quantum expanders were introduced in [2] for their application to quantum spin chains and in [5] for applications to quantum statistical zero knowledge. Here we (following [2] and [4]) define a (N, D, λ) quantum expander to be a quantum operation E that • Has N-dimensional input and output. • Has D Kraus operators. • Has second-largest singular value ≤ λ. Equivalently, if E(ρ) = ρ and tr ρσ = 0 then ‖E(σ)‖2 ≤ λ‖σ‖2,

We give a simple recipe for translating walks on Cayley graphs of a group G into a quantum operation on any irrep of G. Most properties of the classical walk carry over to the quantum operation: degree becomes the number of Kraus operators, the spectral gap becomes the gap of the quantum operation (viewed as a linear map on density matrices), and the quantum operation is efficient whenever the classical walk and the quantum Fourier transform on G are efficient. This means that using classical constant-degree constant-gap families of Cayley expander graphs on e.g. the symmetric group, we can construct efficient families of quantum expanders.