Quantum Interior Point Methods for Semidefinite Optimization

Abstract

We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact search direction and is not guaranteed to explore only feasible points; the second scheme uses a nullspace representation of the Newton linear system to ensure feasibility even with inexact search directions. The second is a novel scheme that might seem impractical in the classical world, but it is well-suited for a hybrid quantum-classical setting. We show that both schemes converge to an optimal solution of the semidefinite optimization problem under standard assumptions. By comparing the theoretical performance of classical and quantum interior point methods with respect to various input parameters, we show that our second scheme obtains a speedup over classical algorithms in terms of the dimension of the problem n, but has worse dependence on other numerical parameters.

Solving the semidefinite relaxation of QUBOs in matrix multiplication time, and faster with a quantum computer

“…QIPMs attempt to speedup the bottleneck of the classical IPM by substituting the classical solution of the Newton linear system with the combined use of QLSA and quantum state tomography (with some classical computation between iterates). Augustino et al [6] present a convergent QIPM for SDO, avoiding the shortcomings prevalent in early works on QIPMs (see, e.g., [39]), by properly symmetrizing the Newton linear system, and utilizing an orthogonal subspace representation of the search directions. This representation guarantees that primal and dual feasibility are satisfied exactly by all the iterates generated by inexact solutions of the Newton linear system obtained via quantum subroutines.…”

“…While this QIPM achieves a speedup in n over the IPM from [34] when m = O(n 2 ), its dependence on κ and ǫ suggest no quantum advantage overall: the complexity of the classical IPM does not depend on κ and its dependence on ǫ −1 is logarithmic. As the authors in [6] note, dependence on the condition number bound κ is particularly problematic in the context of IPMs.…”

Quantum Resources Required to Block-Encode a Matrix of Classical Data

“…The framework has led to faster algorithms for phase estimation [6], quantum gradients [7], and improved Hamiltonian simulation and regression techniques [1], [8]. Furthermore, block-encoding has been used to analyze quantum optimization algorithms [9], [10] and related algorithms for portfolio optimization [11]. Many of these algorithms make use of the quantum singular value transform (QSVT) algorithm [2], [12], which uses a technique known as quantum signal processing (QSP) [8], [13] that performs polynomial transformations on the block-encoded matrix.…”

“…In practice, this is accomplished by loading the sign bit into an ancilla register and applying a Pauli-Z gate, and then unloading the sign bit. 10 The binary tree data structure in Fig. 7 has 2N − 1 nodes.…”

“…However, we notice that in the above construction, all that matters is the 9 Writing the angle as θ 1y ensures that when the subscript is interpreted as an integer written in binary, the angle at step 1 is θ 1 , the angle at step 2 is one of {θ 2 , θ 3 }, the angle at step 3 is one of {θ 4 , θ 5 , θ 6 , θ 7 }, etc. 10 We note that complex amplitudes could also be considered; in that case, one would load the complex phase into an ancilla register and then apply a controlled rotation by that angle about the Z axis.…”