Linear optical circuits with single-photon sources offer a promising platform for quantum chemistry and machine learning. However, current applications are all based on support vector machines or gradient-free optimization methods. This paper develops classical and quantum algorithms for evaluating the analytic gradients of linear optical circuits with respect to their phase parameters. First, we set up a general framework by characterising the class of observables whose expectation values can be estimated efficiently by sampling from a passive linear optical circuit with finitely many photons. We then show how to compute the gradients of the expectation values of a special class of ``non-interacting'' observables arising in full-counting-statistics. Our differentiation algorithm uses the Halmos dilation and requires evaluating two circuits of twice the size, using one additional photon. Building on the methods of full-counting-statistics, we show how to recover the gradients of arbitrary observables from the gradient of a non-interacting characteristic function. Throughout the paper, we compare the performance of classical and quantum algorithms on the same estimation problems, analysing the sampling complexity of the algorithms and suggesting different cases for which quantum speed-ups could be obtained.
Linear optical circuits with single-photon sources offer a promising platform for quantum chemistry and machine learning. However, current applications are all based on support vector machines or gradient-free optimization methods. This paper develops classical and quantum algorithms for evaluating the analytic gradients of linear optical circuits with respect to their phase parameters. First, we set up a general framework by characterising the class of observables whose expectation values can be estimated efficiently by sampling from a passive linear optical circuit with finitely many photons. We then show how to compute the gradients of the expectation values of a special class of ``non-interacting'' observables arising in full-counting-statistics. Our differentiation algorithm uses the Halmos dilation and requires evaluating two circuits of twice the size, using one additional photon. Building on the methods of full-counting-statistics, we show how to recover the gradients of arbitrary observables from the gradient of a non-interacting characteristic function. Throughout the paper, we compare the performance of classical and quantum algorithms on the same estimation problems, analysing the sampling complexity of the algorithms and suggesting different cases for which quantum speed-ups could be obtained.
We present a classical algorithm for approximating the expectation values of observables in linear optical circuits with arbitrary product input states, achieving additive-error accuracy. This result indicates that current applications of photonic systems aimed at demonstrating practical quantum advantage through expectation value estimation, such as photonic variational algorithms, may face challenges in attaining the computational advantage. It also implies the output probabilities of boson sampling with arbitrary product input states can be efficiently approximated by our method, so that boson sampling becomes efficiently simulable when its output probability distribution is polynomially sparse. We also develop an efficient classical algorithm for estimating transition amplitudes of arbitrary product states in linear optical circuits. This provides additive-error approximation algorithms for matrix functions associated with linear optical circuits, such as the (loop-)Hafnian, which are of independent interest. As an application, it solves the generalized molecular vibronic spectra problem [C. Oh et al., Nat. Phys. 20, 225 (2024)10.1038/s41567-023-02308-9], previously suggested as a candidate for practical quantum advantage. Finally, we extend our framework to near-Clifford circuits, enabling classical approximation of their expectation values.
Variational quantum algorithms are a leading approach to near-term quantum computing, but their scalability can be limited by barren plateaus and the sampling cost of resolving small changes in the loss landscape. Here, we study the trainability of passive linear-optical quantum circuits and introduce a framework based on the ratio of sample variance to circuit variance. This ratio determines the number of circuit samples required to resolve local loss differences and gradients to proportional accuracy. We apply this framework to photon-number observables and identify both trainable and non-trainable regimes. Supported by analytic results and a numerically observed polynomial decay of the circuit variance, we find that fixed-order photon-number polynomials require only polynomially many samples as the system size grows, whereas high-order polynomials and observables based on output probabilities generally require exponentially many samples. Within the trainable regime, we further identify classes of observables in which quantum estimation achieves a polynomial speed-up over multiple classical methods. Within this family, neural network observables provide one practical construction that allow measurement outcomes to be efficiently processed into the desired polynomial. These results establish photonic variational quantum computing as a promising platform for near-term applications.
An important application for near-term quantum computing lies in optimization tasks, with applications ranging from quantum chemistry and drug discovery to machine learning. In many settings, most prominently in so-called parametrized or variational algorithms, the objective function is a result of hybrid quantum-classical processing. To optimize the objective, it is useful to have access to exact gradients of quantum circuits with respect to gate parameters. This paper shows how gradients of expectation values of quantum measurements can be estimated using the same, or almost the same, architecture that executes the original circuit. It generalizes previous results for qubit-based platforms, and proposes recipes for the computation of gradients of continuous-variable circuits. Interestingly, in many important instances it is sufficient to run the original quantum circuit twice while shifting a single gate parameter to obtain the corresponding component of the gradient. More general cases can be solved by conditioning a single gate on an ancilla.
Though parameter shift rules have drastically improved gradient estimation methods for several types of quantum circuits, leading to improved performance in downstream tasks, so far they have not been transferable to linear optics with single photons. In this work, we derive an analytical formula for the gradients in these circuits with respect to phaseshifters via a generalized parameter shift rule, where the number of parameter shifts depends linearly on the total number of photons. Experimentally, this enables access to derivatives in photonic systems without the need for finite difference approximations. Building on this, we propose two strategies through which one can reduce the number of shifts in the expression, and hence reduce the overall sample complexity. Numerically, we show that this generalized parameter-shift rule can converge to the minimum of a cost function with fewer parameter update steps than alternative techniques. We anticipate that this method will open up new avenues to solving optimization problems with photonic systems, as well as provide new techniques for the experimental characterization and control of linear optical systems.
Abstract Many near-term quantum computing algorithms are conceived as variational quantum algorithms, in which parameterized quantum circuits are optimized in a hybrid quantum–classical setup. Examples are variational quantum eigensolvers, quantum approximate optimization algorithms as well as various algorithms in the context of quantum-assisted machine learning. 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. We will refer to the way in which one can combine these circuit evaluations as gradient rules. This work provides a comprehensive picture of the family of gradient rules that vary parameters of quantum gates individually. The most prominent known members of this family are the parameter shift rule (PSR) and the finite differences method. To unite this family, we propose a generalized PSR that expresses all members of the aforementioned family as special cases, and discuss how all of these can be seen as providing access to a linear combination of exact first- and second-order derivatives. We further prove that a PSR with one non-shifted evaluation and only one shifted circuit evaluation does not exist, and introduce a novel perspective for approaching new gradient rules.
We develop a backpropagation algorithm for evaluating parameter gradients in quantum circuits using Pauli propagation simulation. The method has computational complexity comparable to that of standard sparse Pauli simulation techniques, while producing gradients whose accuracy is of the same order as the corresponding observable expectation values. By exploiting the reversibility of quantum circuits, the algorithm reduces the memory cost by a factor of O(nparam) compared with conventional reverse-mode automatic differentiation, where nparam denotes the number of parameters in the circuit. Compared with finite difference methods, the algorithm is O(nparam) more efficient in function evaluations. These features enable efficient and accurate classical optimization of quantum circuits for applications such as state preparation and time-evolution compression, while also allowing operator-complexity measures such as the operator stabilizer Rényi entropy to be monitored and regularized during optimization. We demonstrate the method by optimizing low-energy state-preparation circuits for transverse-field Ising models in one, two, and three dimensions and for the three-dimensional Heisenberg model, and by compressing two-dimensional time-evolution circuits.