Quantum Computation as Geometry

Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.

computer: quantum
geometry
quantum circuit
quantum algorithm

References(19)

Figures(1)

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An alternate way of viewing this cost function is as a penalty metric of the kind used in sub-Riemannian geometry (, p. 18)

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The cost function has the property that F(αH) = |α|F(H), where α is any real number. This property and the chain rule imply invariance of the length with respect to reparameterization

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Our metric is superficially similar to the usual metric of Euclidean space, and it is tempting to suppose that geodesics must be straight lines, i.e., constant Hamiltonians. However, the Pauli coefficients for the Hamiltonian actually correspond to a changing local basis for the tangent space, not a fixed basis, and hence constant Hamiltonians do not, in general, give rise to geodesics

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The operator norm of X is defined as \(‖X‖ = max|Ψ⟩|⟨Ψ|X|Ψ⟩| \), where the maximization is over all normalized vectors, \(|⟨Ψ|Ψ⟩|2 = 1 \)

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The overhead factors in this theorem may be substantially improved, e.g., by making use of higher order analyses in lemmas 1 to 3. However, the key point—that U can be accurately approximated with a number of gates that scales polynomially with d(I, U)—remains the same

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We thank

S. Aaronson
,
M. de Burgh
,
J. Dodd
,
C. Hill
,
A. Hines

et al.