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Quantum Information and Computation

Quantum expanders from any classical Cayley graph expander

Aram W. Harrow

Published Sep 1, 2008• 23 citations
Publisher

Abstract

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,

DOI
10.5555/2017011.2017013

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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.
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.
We construct a constant-degree and constant-gap quantum expander on nnn qubits where each unitary can be implemented by a depth-111 and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation S=Θ(Δ−1/2)S = Θ(Δ^{-1/2})S=Θ(Δ−1/2); this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single TTT gate, a single T†T^{\dagger}T† gate, or a depth-111 Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.
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.
A classical t-tensor product expander is a natural way of formalising correlated walks of t particles on a regular expander graph. A quantum t-tensor product expander is a completely positive trace preserving map that is a straightforward analogue of a classical t-tensor product expander. Interest in these maps arises from the fact that iterating a quantum t-tensor product expander gives us a unitary t-design, which has many applications to quantum computation and information. We show that the zigzag product of a high dimensional quantum expander (i.e. t = 1) of moderate degree with a moderate dimensional quantum t-tensor product expander of low degree gives us a high dimensional quantum t-tensor product expander of low degree. Previously such a result was known only for quantum expanders i.e. t = 1. Using the zigzag product we give efficient constructions of quantum t-tensor product expanders in dimension D where t = polylog(D). We then show how replacing the zigzag product by the generalised zigzag product leads to almost-Ramanujan quantum tensor product expanders i.e. having near-optimal almost quadratic tradeoff between the degree and the second largest singular value. Both the products give better tradeoffs between the degree and second largest singular value than what was previously known for efficient constructions.
We introduce the concept of quantum tensor product expanders. These generalize the concept of quantum expanders, which are quantum maps that are efficient randomizers and use only a small number of Kraus operators. Quantum tensor product expanders act on several copies of a given system, where the Kraus operators are tensor products of the Kraus operator on a single system. We begin with the classical case, and show that a classical two-copy expander can be used to produce a quantum expander. We then discuss the quantum case and give applications to the Solovay-Kitaev problem. We give probabilistic constructions in both classical and quantum cases, giving tight bounds on the expectation value of the largest nontrivial eigenvalue in the quantum case.