Demonstration of Hardware Efficient Photonic Variational Quantum Algorithm

TL;DR

The paper demonstrates a hardware-efficient variational quantum algorithm implemented on a four-mode photonic integrated circuit to solve a factorization instance with N=35. It encodes the problem as the ground-state search of an Ising Hamiltonian Hp and uses a photonic ansatz U(θ,φ) optimized by a classical gradient-descent loop to minimize E(θ,φ)=⟨ψ0|U†HpU|ψ0⟩. Experimental results show convergence to the ground state corresponding to the factor pair 5 and 7, with ground-state degeneracy yielding two valid encodings and high success probability across many randomized initializations. This work highlights the viability of photonic VQAs on current hardware, discusses scalability and limitations related to mode counts and probabilistic gates, and points to broader applicability in problems such as molecular ground-state estimation and graph optimization.

Abstract

Quantum computing has brought a paradigm change in computer science, where non-classical technologies have promised to outperform their classical counterpart. Such an advantage was only demonstrated for tasks without practical applications, still out of reach for the state-of-art quantum technologies. In this context, a promising strategy to find practical use of quantum computers is to exploit hybrid quantum-classical models, where a quantum device estimates a hard-to-compute quantity, while a classical optimizer trains the parameters of the model. In this work, we demonstrate that single photons and linear optical networks are sufficient for implementing Variational Quantum Algorithms, when the problem specification, or ansatz, is tailored to this specific platform. We show this by a proof-of-principle demonstration of a variational approach to tackle an instance of a factorization task, whose solution is encoded in the ground state of a suitable Hamiltonian. This work which combines Variational Quantum Algorithms with hardware efficient ansatzes for linear-optics networks showcases a promising pathway towards practical applications for photonic quantum platforms.

Figures (5)

• Figure 1: Variational Quantum Algorithm to solve a factorization task. a) In the noisy intermediate scale quantum (NISQ) era, before fully fault-tolerant large-scale quantum computers are available, Variational Quantum Algorithms (VQAs) provide a fruitful toolset to tackle a broad variety of problems, whose solution rely on quantum hardware to estimate quantities that are hard to compute and a classical optimization process to find the optimal parameters for the model. b) The prime factorization problem constitutes one of the corner stones of modern cryptography. Though quantum algorithms are known to factor prime numbers exponentially faster than classically possible, their implementation requires quantum devices of currently infeasible scale and quality. c) We apply a variational method to the factorization problem, tailoring the algorithm to a quantum photonic platform.
• Figure 2: Conceptual scheme of the proposed single-photon linear-optics based factorization scheme. The process is hybrid quantum (above) and classical (below) architecture. The quantum part consists of a tunable photonic circuit (PIC) with four input/output modes Clements:16. This device contains tunable Mach-Zehnder interferometers ($L=6$), with two tunable parameters, $\theta$ and $\phi$, for enabling arbitrary unitary evolutions. The circuit rotates the input state $|\psi_0\rangle$ to $U(\vartheta,\varphi)|\psi_0\rangle$, which is then measured in the computational basis. The classical part covers the calculation of a cost function $\mathcal{E}(\vartheta,\varphi)$ related to the expectation value of the Hamiltonian whose ground state provides the solution to a computational problem (here, prime factorization of $N=35$). Based on the measurement outcome of $U(\vartheta,\varphi)|\psi_0\rangle$, a classical gradient descent algorithm is used to adapt $U(\vartheta,\varphi)$, minimizing the cost function $\mathcal{E}(\vartheta,\varphi)$. The gradients are computed by the forward difference method which involves $2L + 1$ times of function evaluations for each iteration of the optimization process.
• Figure 3: Experimental setup. Orthogonally polarized single-photon pairs generated by a type II Spontaneous Parametric Down-Conversion (SPDC) process are separated using a fiber polarizing beam-splitter (PBS), where one output port is used to herald a pair creation event. The remaining photon is injected into the photonic integrated circuit (PIC) in a fixed spatial mode representing the input state $\ket{\psi_0}=\ket{00}$. The PIC implements a unitary $U(\vartheta, \varphi)$ on the photon-path degree-of-freedom, creating the trial state $\ket{\psi(\vartheta, \varphi)} = U(\vartheta, \varphi) \ket{\psi_0}$. This state is measured in the two-qubit computational basis, where the basis states $\ket{00}$ ($\ket{01}$) are identified by a photon being present in the first (second) output mode, and so on. From the acquired output statistics, the trial states energy $\mathcal{E}(\vartheta, \varphi) = \braket{\psi(\vartheta, \varphi)|H_p|\psi(\vartheta, \varphi)}$ with respect to the Hamiltonian $H_p$ of Eq. \ref{['optHamiltmain']} can be computed. A classical gradient descent algorithm is employed to minimize $\mathcal{E}(\vartheta, \varphi)$ by varying the internal and external phases $\vartheta, \varphi$.
• Figure 4: Convergence of the estimated ground state energy for different initial configurations. Calculated energy $\braket{\Psi | H_p | \Psi}$ (Eq. \ref{['eq:energy_unitary1']}) at each optimization step, for $117$ different initial configurations of the experiment (colored lines), as they approach the theoretically predicted ground state energy $\mathrm{E_g}$ (dashed line). The configuration at each step consists in the set of internal and external phases $(\vartheta, \varphi)$ applied to the integrated circuit, which amounts to a unitary operation $U(\vartheta, \varphi)$. Accordingly, at the $i$th step, the output state will be $|\Psi_i\rangle=U(\vartheta_i, \varphi_i)|\psi_0\rangle$, where $|\psi_0\rangle$ is the initial input. To evaluate the obtained energy, due to the form of our Hamiltonian $H_p$ (see Eq. \ref{['optHamiltmain']}), it is sufficient to post-process the statistics obtained by projecting the output state onto the computational basis. The black line shows the mean experimental energy at every step, where experiments with a final energy $\mathcal{E} \geq 0$ have been removed (that is excluding 5.0 out of 117.0 repetitions). A magnified region close to $\mathrm{E_g}$ is depicted in the inset. The theoretically predicted ground state energy is reached by the optimization procedure after 27, 28, 115 steps within a relative uncertainty of 1%, 0.1%, and 0.01% respectively. Let us note that the achieved energy values are lower than 0, because we are removing a constant term from the Eq. \ref{['optHamiltmain']}.
• Figure 5: Experimental factorization results.a) The bars correspond to the experimental frequencies with which a photon is detected at each of the four outputs of our circuit. The reported values are obtained by averaging over the results of 117.0 different experimental optimizations, starting from different configurations. For our encoding, injecting one photon in the $i$-th mode constitutes the $i$-th element of a 2-qubits computational basis. Therefore, the bars representing the highest frequencies indicate the ground state $\ket{x_1 y_1}$ of the Hamiltonian outlined in Eq. \ref{['optHamiltmain']}, which is an almost equal superposition of $\ket{01}$ and $\ket{10}$. This effectively resolves the factorization problem, allowing us to construct the factors as $1 x_1 1 = 101 =5$ and $1 y_1 1 = 111 =7$, or vice versa. b) As all states of the form $\sqrt{\alpha}\ket{01} + \sqrt{1-\alpha}\ket{10}$ exhibit the same energy with respect to $H_p$ (see Supplemental Material section IV), every individual optimization converges to one arbitrary of those possible outcomes. The average output of independent repetitions thus approaches the equal superposition $\alpha=0.5$, as can be seen on the fidelity of $0.97546(5)$ towards this state after 117 repetitions (violet), while the fidelities to both $\ket{01}$ (green) and $\ket{10}$ (teal) concordantly revolve around $0.7$. The fidelity (Eq. \ref{['equ:fidelity']}) is evaluated by interpreting the output distribution measured in the computational basis as normalized state vector. For both plots, the uncertainties (shaded regions for 2 standard deviations), evaluated considering an underlying Poissonian statistics, using a Monte Carlo Simulation and are too small to be visible.