Generalized quantum circuit differentiation rules

• URL: http://arxiv.org/abs/2108.01218v2
• Date: Sun, 17 Oct 2021 11:52:36 GMT
• Title: Generalized quantum circuit differentiation rules
• Authors: Oleksandr Kyriienko, Vincent E. Elfving
• Abstract summary: Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits. Here, we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators.
• Score: 23.87373187143897
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Variational quantum algorithms that are used for quantum machine learning rely on the ability to automatically differentiate parametrized quantum circuits with respect to underlying parameters. Here, we propose the rules for differentiating quantum circuits (unitaries) with arbitrary generators. Unlike the standard parameter shift rule valid for unitaries generated by operators with spectra limited to at most two unique eigenvalues (represented by involutory and idempotent operators), our approach also works for generators with a generic non-degenerate spectrum. Based on a spectral decomposition, we derive a simple recipe that allows explicit derivative evaluation. The derivative corresponds to the weighted sum of measured expectations for circuits with shifted parameters. The number of function evaluations is equal to the number of unique positive non-zero spectral gaps (eigenvalue differences) for the generator. We apply the approach to relevant examples of two-qubit gates, among others showing that the fSim gate can be differentiated using four measurements. Additionally, we present generalized differentiation rules for the case of Pauli string generators, based on distinct shifts (here named as the triangulation approach), and analyse the variance for derivative measurements in different scenarios. Our work offers a toolbox for the efficient hardware-oriented differentiation needed for circuit optimization and operator-based derivative representation.

Related papers

• Variational approach to photonic quantum circuits via the parameter shift rule [0.0]
  We derive a formulation of the parameter shift rule for reconfigurable optical linear circuits based on the Boson Sampling paradigm. We also present similar rules for the computations of integrals over the variational parameters. We employ the developed approach to experimentally test variational algorithms with single-photon states processed in a reconfigurable 6-mode universal integrated interferometer.
  arXiv  Detail & Related papers  (2024-10-09T15:06:17Z)
• Geometric Quantum Machine Learning with Horizontal Quantum Gates [41.912613724593875]
  We propose an alternative paradigm for the symmetry-informed construction of variational quantum circuits. We achieve this by introducing horizontal quantum gates, which only transform the state with respect to the directions to those of the symmetry. For a particular subclass of horizontal gates based on symmetric spaces, we can obtain efficient circuit decompositions for our gates through the KAK theorem.
  arXiv  Detail & Related papers  (2024-06-06T18:04:39Z)
• Characterizing randomness in parameterized quantum circuits through expressibility and average entanglement [39.58317527488534]
  Quantum Circuits (PQCs) are still not fully understood outside the scope of their principal application. We analyse the generation of random states in PQCs under restrictions on the qubits connectivities. We place a connection between how steep is the increase on the uniformity of the distribution of the generated states and the generation of entanglement.
  arXiv  Detail & Related papers  (2024-05-03T17:32:55Z)
• Majorization-based benchmark of the complexity of quantum processors [105.54048699217668]
  We numerically simulate and characterize the operation of various quantum processors. We identify and assess quantum complexity by comparing the performance of each device against benchmark lines. We find that the majorization-based benchmark holds as long as the circuits' output states have, on average, high purity.
  arXiv  Detail & Related papers  (2023-04-10T23:01:10Z)
• General quantum algorithms for Hamiltonian simulation with applications to a non-Abelian lattice gauge theory [44.99833362998488]
  We introduce quantum algorithms that can efficiently simulate certain classes of interactions consisting of correlated changes in multiple quantum numbers. The lattice gauge theory studied is the SU(2) gauge theory in 1+1 dimensions coupled to one flavor of staggered fermions. The algorithms are shown to be applicable to higher-dimensional theories as well as to other Abelian and non-Abelian gauge theories.
  arXiv  Detail & Related papers  (2022-12-28T18:56:25Z)
• General parameter-shift rules for quantum gradients [0.03823356975862005]
  Variational quantum algorithms are ubiquitous in applications of noisy intermediate-scale quantum computers. We show that a general parameter-shift rule can reduce the number of circuit evaluations significantly. Our approach additionally reproduces reconstructions of the evaluated function up to a chosen order, leading to known generalizations of the Rotosolve algorithm.
  arXiv  Detail & Related papers  (2021-07-26T18:00:02Z)
• Analytic gradients in variational quantum algorithms: Algebraic extensions of the parameter-shift rule to general unitary transformations [0.0]
  We propose several extensions of the parametric-shift-rule to formulating gradients as linear combinations of expectation values for generators with general eigen-spectrum. Our approaches are exact and do not use any auxiliary qubits, instead they rely on a generator eigen-spectrum analysis.
  arXiv  Detail & Related papers  (2021-07-16T21:37:31Z)
• Accurate methods for the analysis of strong-drive effects in parametric gates [94.70553167084388]
  We show how to efficiently extract gate parameters using exact numerics and a perturbative analytical approach. We identify optimal regimes of operation for different types of gates including $i$SWAP, controlled-Z, and CNOT.
  arXiv  Detail & Related papers  (2021-07-06T02:02:54Z)
• Single-component gradient rules for variational quantum algorithms [1.3047205680129093]
  A common bottleneck of any such algorithm is constituted by the optimization of the variational parameters. A popular set of optimization methods work on the estimate of the gradient, obtained by means of circuit evaluations. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually.
  arXiv  Detail & Related papers  (2021-06-02T18:00:10Z)
• Coherent randomized benchmarking [68.8204255655161]
  We show that superpositions of different random sequences rather than independent samples are used. We show that this leads to a uniform and simple protocol with significant advantages with respect to gates that can be benchmarked.
  arXiv  Detail & Related papers  (2020-10-26T18:00:34Z)
• Estimating the gradient and higher-order derivatives on quantum hardware [1.2891210250935146]
  We show how arbitrary-order derivatives can be analytically evaluated in terms of simple parameter-shift rules. We also consider the impact of statistical noise by studying the mean squared error of different derivative estimators.
  arXiv  Detail & Related papers  (2020-08-14T18:00:10Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.