Variational measurement-based quantum computation for generative modeling

TL;DR

This work treats the intrinsic randomness of measurement-based quantum computation (MBQC) as a resource for generative modeling, introducing a variational MBQC framework that controls byproduct randomness via trainable correction probabilities. The authors formulate two MBQC-based generative models—a mixed-unitary channel $ ext{E}_c$ and its corrected variant $ ilde{ ext{E}}_c$—and compare them to a unitary MBQC model $U_c$, showing both theoretical expressivity advantages and empirical performance gains in learning tasks. They employ a maximum mean discrepancy (MMD) implicit loss and derive gradients for variational angles and correction probabilities, enabling training from samples. Across learning tasks (mixed-unitary distributions and double Gaussians) and under noise, the results indicate that embracing MBQC randomness can enhance expressivity and learning performance, motivating further MBQC-based approaches for quantum generative modeling.

Abstract

Measurement-based quantum computation (MBQC) offers a fundamentally unique paradigm to design quantum algorithms. Indeed, due to the inherent randomness of quantum measurements, the natural operations in MBQC are not deterministic and unitary, but are rather augmented with probabilistic byproducts. Yet, the main algorithmic use of MBQC so far has been to completely counteract this probabilistic nature in order to simulate unitary computations expressed in the circuit model. In this work, we propose designing MBQC algorithms that embrace this inherent randomness and treat the random byproducts in MBQC as a resource for computation. As a natural application where randomness can be beneficial, we consider generative modeling, a task in machine learning centered around generating complex probability distributions. To address this task, we propose a variational MBQC algorithm equipped with control parameters that allow one to directly adjust the degree of randomness to be admitted in the computation. Our algebraic and numerical findings indicate that this additional randomness can lead to significant gains in expressivity and learning performance for certain generative modeling tasks, respectively. These results highlight the potential advantages in exploiting the inherent randomness of MBQC and motivate further research into MBQC-based algorithms.

Figures (10)

• Figure 1: Variational measurement-based quantum computation architecture. Depicted is the $N$-qubit circuit model picture for the variational MBQC proposed in this paper. Here, MBQC can be understood as $D$ alternating layers of local $R_Z$-rotations (parametrized by $\{\theta^j_i\}_{i,j}$), byproduct operators $Z^{\tilde{s}^j_i}$ and Clifford Quantum Cellular Automata $T_c$ (C-QCA) as exemplified in Fig. \ref{['fig:glider']}. In this approach to generative modeling, we learn the variational angles $\{\theta^j_i\}_{i,j}$ and probabilities $\{p^j_i\}_{i,j}$ for correcting byproducts. To better control the impact of the randomness on the computation, we also maintain conditional control of Pauli operators at the end of the computation on the appearance of byproducts by $P_{D+1}(\bm{\tilde{s}})$ [see Eq. \ref{['eq:mbqc_byproducts']}] as in standard adaptive MBQC. As shown in Sec. \ref{['sec:two_models_compare']}, this significantly improves the expressivity and versatility of the learning model. The resulting mixed unitary channel is given by Eq. \ref{['eq:mbqc_channel_model']}. An implicit loss is calculated classically by comparing measurement samples from the MBQC with samples from the real distribution.
• Figure 2: Circuit picture of an MBQC based on C-QCA. An MBQC on the cluster state (top) which only utilizes measurements in the $XY$-plane can be understood in the circuit picture (bottom) where local $Z$-rotations are interleaved with C-QCAs $T_c$. The output qubits in the third column (white) are measured in the $Z$-basis. Measurements on a cluster state are inherently random. In this example the bottom left qubit of the cluster state is measured in the $e^{i\theta_3^1 Z}\left| \pm \right>\left< \pm \right|e^{-i\theta_3^1 Z}$-basis, yielding the outcome $+1$ and thus showing no $Z$-byproduct in the circuit. We also depict two qubits (pink, top left and center) for which the measurements yielded an outcome $-1$, resulting in $Z$-byproducts in the equivalent circuit. These byproducts can be corrected to make the computation unitary, but we will implement such corrections only with a certain probability.
• Figure 3: C-QCA and byproduct propagation. The C-QCA consists of a layer of controlled-phase gates between neighboring qubits and a layer of Hadamard gates on each qubit (assuming periodic boundary conditions). Byproduct propagation is exemplified for Pauli-$X$ and -$Z$ operators, and can be straightforwardly extended to any product of Pauli operators (see Ref. nautrup2023measurement). The rule for byproduct propagation can be understood by the transition function in Eq. \ref{['eq:T-func_3']}.
• Figure 4: Variational MBQC learning performance across various system sizes. Two learning models, based on $\tilde{\mathcal{E}}_c(\bm{\theta},\bm{p})$ (blue) in Eq. \ref{['eq:mbqc_channel_model']} and $U_c(\bm{\theta})$ (orange) in Eq. \ref{['eq:mbqc_unitary']}, respectively, are trained to approximate a given target distribution, generated from a mixed unitary channel based on $\tilde{\mathcal{E}}_c$ in Eq. \ref{['eq:mbqc_channel_model']}. All models use the same number of qubits and layers. The plot shows the mean loss across $100$ epochs averaged over 12 randomly initialized iterations. The shaded areas correspond to the respective standard deviations. Solid lines: Learning performance of $U_c$ (orange) and $\tilde{\mathcal{E}}_c$ (blue), respectively, for $N=8$ qubits and depth $D=7$. Dashed lines: Learning performance of $U_c$ (orange) and $\tilde{\mathcal{E}}_c$ (blue), respectively, for $N=5$ qubits and depth $D=4$. This plot shows the improved performance of the mixed unitary models over the unitary models in this task.
• Figure 5: Variational MBQC learning performance on a double Gaussian across various system sizes. Two learning models, based on $\tilde{\mathcal{E}}_c(\bm{\theta},\bm{p})$ (blue) in Eq. \ref{['eq:mbqc_channel_model']} and $U_c(\bm{\theta})$ (orange) in Eq. \ref{['eq:mbqc_unitary']}, respectively, are trained to approximate a given target distribution, generated from a double Gaussian distribution in Eq. \ref{['eq:double_gauss']}. The plot shows the mean loss over 200 episodes averaged over eight randomly initialized iterations. The shaded areas correspond to the respective standard deviations. Solid lines: Learning performance of $U_c$ (orange) and $\tilde{\mathcal{E}}_c$ (blue), respectively, for $N=8$ qubits and depth $D=7$. Dashed lines: Learning performance of $U_c$ (orange) and $\tilde{\mathcal{E}}_c$ (blue), respectively, for $N=5$ qubits and depth $D=4$. This plot shows the improved performance of the mixed unitary models over the unitary models in this task, with the gap widening as the system size increases.

Theorems & Definitions (1)

• Theorem 1