© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:1808.10521

Efficient quantum tensor product expanders and unitary t-designs via the zigzag product

Sen, Pranab

Published Aug 30, 2018• 6 citations

Abstract

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.

DOI: 10.48550/arxiv.1808.10521

Related papers

Classical and quantum tensor product expanders

M. B. Hastings, Aram W. Harrow

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.

Quantum Information and Computation•Mar 1, 2009•46 citations

Quantum expanders from any classical Cayley graph expander

Aram W. Harrow

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,

Quantum Information and Computation•Sep 1, 2008•23 citations

Classical and Quantum Tensor Product Expanders

M. B. Hastings, Aram W. Harrow

We introduce the concept of quantum tensor product expanders. These are expanders that 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.

arXiv:0804.0011•Apr 1, 2008•0 citations

Optimal Quantum (Tensor Product) Expanders From Unitary Designs

Cécilia Lancien

Abstract In this work we investigate how quantum expanders (i.e., quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a $2$-design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form $U^{\otimes k}$, with $U$ sampled from a $2k$-design measure, then the corresponding random quantum channel is typically an optimal $k$-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).

arXiv:2409.17971•International Mathematics Research Notices•Dec 30, 2025•1 citation

Efficient quantum circuits for high-dimensional representations of SU(n) and Ramanujan quantum expanders

Vishnu Iyer, Siddhartha Jain, Stephen Jordan +1 more

We present efficient quantum circuits that implement high-dimensional unitary irreducible representations (irreps) of $SU(n)$, where $n \ge 2$ is constant. For dimension $N$ and error $ε$, the number of quantum gates in our circuits is polynomial in $\log(N)$ and $\log(1/ε)$. Our construction relies on the Jordan-Schwinger representation, which allows us to realize irreps of $SU(n)$ in the Hilbert space of $n$ quantum harmonic oscillators. Together with a recent efficient quantum Hermite transform, which allows us to map the computational basis states to the eigenstates of the quantum harmonic oscillator, this allows us to implement these irreps efficiently. Our quantum circuits can be used to construct explicit Ramanujan quantum expanders, a longstanding open problem. They can also be used to fast-forward the evolution of certain quantum systems.

quant-pharXiv:2602.15180•Feb 18, 2026•0 citations

Depth-1 expanders on the unitary group and applications

Anurag Anshu, Shankar Balasubramanian, Jonas Haferkamp +2 more

We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ 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 $S = Θ(Δ^{-1/2})$; 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 $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ 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.

quant-pharXiv:2609.01605•Sep 2, 2026•0 citations

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.

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 introduce the concept of quantum tensor product expanders. These are expanders that 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.

Abstract In this work we investigate how quantum expanders (i.e., quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a

2

-design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form

U^{\otimes k}

, with

U

sampled from a

2k

-design measure, then the corresponding random quantum channel is typically an optimal

k

-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).

We present efficient quantum circuits that implement high-dimensional unitary irreducible representations (irreps) of

SU(n)

, where

n \ge 2

is constant. For dimension

N

and error

ε

, the number of quantum gates in our circuits is polynomial in

\log(N)

and

\log(1/ε)

. Our construction relies on the Jordan-Schwinger representation, which allows us to realize irreps of

SU(n)

in the Hilbert space of

n

quantum harmonic oscillators. Together with a recent efficient quantum Hermite transform, which allows us to map the computational basis states to the eigenstates of the quantum harmonic oscillator, this allows us to implement these irreps efficiently. Our quantum circuits can be used to construct explicit Ramanujan quantum expanders, a longstanding open problem. They can also be used to fast-forward the evolution of certain quantum systems.

We construct a constant-degree and constant-gap quantum expander on

n

qubits where each unitary can be implemented by a depth-

1

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})

; 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

T

gate, a single

T^{\dagger}

gate, or a depth-

1

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.