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International Journal of Foundations of Computer Science

APPLICATIONS OF FINSLER GEOMETRY TO SPEED LIMITS TO QUANTUM INFORMATION PROCESSING

Benjamin Russell, Susan Stepney

Published Jun 1, 2014• 15 citations
Publisher

Abstract

We are interested in fundamental limits to computation imposed by physical constraints. In particular, the physical laws of motion constrain the speed at which a computer can transition between well-defined states. Here, we discuss speed limits in the context of quantum computing. We review some relevant parts of the theory of Finsler metrics on Lie groups and homogeneous spaces such as the special unitary groups and complex projective spaces. We show how these constructions can be applied to analysing the limit to the speed of quantum information processing operations in constrained quantum systems with finite dimensional Hilbert spaces of states. We demonstrate the approach applied to a spin chain system.

DOI
10.1142/s0129054114400073

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We are interested in fundamental limits to computation imposed by physical constraints. In particular, the physical laws of motion constrain the speed at which a computer can transition between well-defined states. Here, we discuss speed limits in the context of quantum computing. We review some relevant parts of the theory of Finsler metrics on Lie groups and homogeneous spaces such as the special unitary groups and complex projective spaces. We show how these constructions can be applied to analysing the limit to the speed of quantum information processing operations in constrained quantum systems with finite dimensional Hilbert spaces of states. We demonstrate the approach applied to a spin chain system.
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We use a specific geometric method to determine speed limits for the implementation of quantum gates in controlled quantum systems that have a specific class of constrained control functions. We achieve this by applying a recent theorem of Shen, which provides a connection between time optimal navigation on Riemannian manifolds and the geodesics of a certain Finsler metric of Randers type. We use the lengths of these geodesics to derive the optimal implementation times (under the assumption of constant control fields) for an arbitrary quantum operation (on a finite dimensional Hilbert space), and explicitly calculate the result for the case of a controlled single spin system in a magnetic field and a swap gate in a Heisenberg spin chain.
We study the maximum speed of quantum computation and how it is affected by limitations on physical resources. We show how the resulting concepts generalize to a broader class of physical models of computation within dynamical systems and introduce a specific algebraic structure representing these speed limits. We derive a family of quantum speed limit results in resource-constrained quantum systems with pure states and a finite dimensional state space, by using a geometric method based on right invariant action functionals on [Formula: see text]. We show that when the action functional is bi-invariant, the minimum time for implementing any quantum gate using a potentially time-dependent Hamiltonian is equal to the minimum time when using a constant Hamiltonian, thus constant Hamiltonians are time optimal for these constraints. We give an explicit formula for the time in these cases, in terms of the resource constraint. We show how our method produces a rich family of speed limit results, of which the generalized Margolus–Levitin theorem and the Mandelstam–Tamm inequality are special cases. We discuss the broader context of geometric approaches to speed limits in physical computation, including the way geometric approaches to quantum speed limits are a model for physical speed limits to computation arising from a limited resource.
Quantum speed limits (QSLs) provide an upper bound for the speed of evolution of quantum states in any physical process. Based on the Stratonovich-Weyl correspondence, we derive a universal QSL bound in arbitrary phase spaces that is applicable for both continuous variable systems and finite-dimensional discrete quantum systems. This QSL bound allows the determination of speed limit bounds in specific phase spaces that are tighter than those in Wigner phase space or Hilbert space under the same metric, as illustrated by several typical examples, e.g., a single-mode free field and NNN-level quantum systems in phase spaces. This QSL bound also provides an experimentally realizable way to examine the speed limit in phase spaces relevant to applications in quantum information and quantum optics.
The speed of quantum evolution is limited under finite energy resources. While most quantum speed limits (QSLs) are formulated in terms of quantum states, they can be extended to the evolution operator itself, and thus impose fundamental limits on how quickly logical gate operations can be implemented on a quantum computer. Here, we derive a general, tight QSL that holds for any unitary evolution under the constraint that the spectral width of the Hamiltonian is bounded. We apply this result to obtain QSLs for several standard quantum gates, including Hadamard, CNOT, and Toffoli gates, finding that the QSL can vary significantly across different gates, including ones with the same entangling power. These findings can be understood geometrically using the Space Curve Quantum Control formalism, which maps unitary evolution to space curves in Euclidean space. In this formalism, the problem of finding QSLs is recast as the problem of finding minimal-length curves obeying a curvature bound. We find that time-optimal gates map to helices of varying dimensions, and that QSLs can be understood from the perspective of a bottleneck principle in which the operator that evolves the slowest governs the minimal gate time.