Sketching the Best Approximate Quantum Compiling Problem

Abstract

This paper considers the problem of quantum compilation from an optimization perspective by fixing a circuit structure of CNOTs and rotation gates then optimizing over the rotation angles. We solve the optimization problem classically and consider algorithmic tools to scale it to higher numbers of qubits. We investigate stochastic gradient descent and two sketchand-solve algorithms. For all three algorithms, we compute the gradient efficiently using matrix-vector instead of matrix-matrix computations. Allowing for a runtime on the order of one hour, our implementation using either sketch-and-solve algorithm is able to compile 9 qubit, 27 CNOT circuits; 12 qubit, 24 CNOT circuits; and 15 qubit, 15 CNOT circuits. Without our algorithmic tools, standard optimization does not scale beyond 9 qubit, 9 CNOT circuits, and, beyond that, is theoretically dominated by barren plateaus.

Escaping barren plateaus in approximate quantum compiling

“…(1) L (α) ≈ 0, all the flip terms are switched off and we arrive at the desired solution V (θ) |0 ≈ |ψ 0 . In A we discuss this weighting scheme in more detail, as well as our implementation of a fast-gradient approach based on [37] which allows us to do experiments with up to 24 qubits. We also developed a number of techniques to improve performance, in particular we use a "surrogate model" which speeds up the calculation of the gradient -we discuss these aspects of the algorithm in B.…”

“…can enhance the optimisation procedure due to the reduced noise in the gradient profile. To increase speed, we apply a fast gradient approach based on [37] by using a more economical matrix-vector multiplication. However this approach must be modified in order to apply it to our case; the main bottleneck is that the gradient of every term in the sum in equation ( 31) should be computed separately and then all of them summed up.…”

“…It is straightforward to show that the following normalized β's, β 2 = 1, maximize the projection: β 0 ∼ 0| V † (θ) |ψ 0 and β j ∼ 0| X j V † (θ) |ψ 0 1 . On every iteration, we compute the latter dot products for β's and normalize the vector β, then substitute β-parameters in the last term in (37) and compute the composite state β * 0 0| + n j=1 β * j 0| X j . Afterwards, we proceed with optimization over the parameters θ as usually.…”

“…For the additional speed up we compute the state V † (θ) |ψ 0 at the beginning of each optimization iteration given the current parameters θ and use it for computation of both, the objective function and its gradient, see [37] for details. This substantially reduces the processing time.…”

“…Really, let us define y 0 = 0| V † (θ) |ψ 0 and y j = 0| X j V † (θ) |ψ 0 . Then the last term in(37) takes the form of squared module of a dot product:| n i=0 β * i y i | 2, which is maximized when β i ∼ y i .…”