Mitigating the barren plateau problem in linear optics

TL;DR

This work addresses barren plateaus in variational quantum algorithms implemented with linear optics by replacing standard phase shifters with dual-valued phase shifters (DVPS) that have two eigenvalues. This reduction collapses the cost landscape harmonics from up to $n$ to a single frequency, enabling gradient-free optimisation via Rotosolve and simplifying gradient evaluation to two evaluations per parameter; the authors present three DVPS implementations: deterministic non-linear optics, measurement-induced non-linearities, and fermion sampling with entangled resources. Empirical results show that DVPS- and fermion-based designs yield smoother cost landscapes and outperform traditional boson-sampling variational algorithms for constrained and unconstrained QUBO problems, with fermion sampling offering additional practical advantages due to classical simulability. The findings highlight a path to faster, quantum-effect-driven optimisation in linear optics while clarifying the practical quantum advantage landscape, and they open questions about extending DVPS concepts to other variational paradigms and their impact on quantum versus classical performance.

Abstract

We demonstrate a significant speedup of variational quantum algorithms that use discrete variable boson sampling when the parametrised phase shifters are constrained to have two distinct eigenvalues. This results in a cost landscape with less local minima and barren plateaus regardless of the problem, ansatz or circuit layout. This works without reliance on any classical pre-processing and allows for the fast gradient-free Rotosolve algorithm to be used. We propose three ways to achieve this by using either non-linear optics, measurement-induced non-linearities, or entangled resource states simulating fermionic statistics. The latter two require linear optics only, allowing for implementation with widely-available technology today. We show this outperforms the best-known boson sampling variational algorithm for all tests we conducted.

Figures (15)

• Figure 1: (a) The boson sampling variational quantum algorithm of Ref. bradler2021certain consists of a linear optical interferometer encoding a parametrised unitary $U(\boldsymbol{\theta})$ and photo-detectors. Here we show an example with a photon inserted into the top two modes on the left, represented by the solid circles, where the stars represent detector clicks which maps to a bit string. The classical computer calculates the cost function $E(\boldsymbol{\theta})$ from multiple shots of this and then updates the parameters in order to optimise this.(b) Each cross-over point corresponds to a Mach-Zehnder interferometer consisting of two phase shifters $\theta,\phi \in [0,2\pi)$ and two fixed 50:50 beamsplitters.
• Figure 2: Rotosolve rotosolve1rotosolve2rotosolve3rotosolve4
• Figure 3: (a) A non-deterministic DVPS consists of one logical mode and two ancillary modes. The logical mode interacts with the lower ancillary mode via a cross-Kerr non-linearity of $\pi$, shown by the square. The ancillary modes interact amongst themselves with a phase shifter of phase $x/2$, a tuneable beamsplitter of angle $x \in [0,2\pi)$, and a second 50:50 beamsplitter. The success of the gate is heralded by the ancillary output state of $|1,0\rangle$. This gate has a success rate of $1/2$. (b) If the non-deterministic gate fails, as heralded by no photon in the upper ancillary mode, then the ancillary photon exits the lower ancillary mode and is rerouted into the input of a second iteration of the non-deterministic gate with double the parameter. We repeat until success.
• Figure 4: (a) A measurement-induced non-linear mapping $|\psi_\text{in}\rangle \mapsto |\psi_\text{out}\rangle$ is obtained by interfering the input state with an ancillary state $|\mathbf{a}\rangle$ in an $(N+1)$-mode linear interferometer and postselecting on the ancillary output. (b) The maximum probability of the non-deterministic dual-valued phase shifter obtained by solving Eq. \ref{['eq:non_linearity_condition']} for the subspace of at most two photons. This uses the circuit of (a) with $N= 2$ ancillary modes and $|\mathbf{a}\rangle = |1,0\rangle$. For $x = 0,\pi$ and $2\pi$, $\mathrm{max}(p_x) = 1$.
• Figure 5: (a) A fermion sampling experiment with two input fermions in the state $|\psi_\mathrm{F}\rangle = a^\dagger_1 a^\dagger_2 |0\rangle$, represented by the solid circles, passed into a linear interferometer encoding some unitary. The clicks of the detectors map to a bit string. (b) The photonic simulation of this consists of a state preparation that transforms the state $a_{11}^\dagger a_{22}^\dagger |0\rangle$, shown by the solid circles, into the entangled state $|\psi_\mathrm{B}\rangle = \frac{1}{\sqrt{2}}(a^\dagger_{11} a_{22}^\dagger - a_{12}^\dagger a_{21}^\dagger)|0\rangle$, where each pair of like-coloured circles represents a term in the superposition. The CNOT gate is constructed from linear optics using the unheralded design of Ref. PhysRevA.65.062324 which has a success probability of $1/9$. This is then inserted into a pair of identical interferometers encoding the same unitary. The superimposed detector click distribution of both interferometers gives rise to the output bit string.

Theorems & Definitions (2)

• Theorem 1: Measurement-induced non-linearities PhysRevA.68.032310
• proof