Minimizing classical resources in variational measurement-based quantum computation for generative modeling

Arunava Majumder; Hendrik Poulsen Nautrup; Hans J. Briegel

quant-phApr 13, 2026

Minimizing classical resources in variational measurement-based quantum computation for generative modeling

Authors: Arunava Majumder, Hendrik Poulsen Nautrup, Hans J. Briegel

Organizations: University of Innsbruck, Department of Theoretical Physics, Technikerstraße 21a, A-6020 Innsbruck, Austria.

Abstract

Measurement-based quantum computation (MBQC) is a framework for quantum information processing in which a computational task is carried out through one-qubit measurements on a highly entangled resource state. Due to the indeterminacy of the outcomes of a quantum measurement, the random outcomes of these operations, if not corrected, yield a variational quantum channel family. Traditionally, this randomness is corrected through classical processing in order to ensure deterministic unitary computations. Recently, variational measurement-based quantum computation (VMBQC) has been introduced to exploit this measurement-induced randomness to gain an advantage in generative modeling. A limitation of this approach is that the corresponding channel model has twice as many parameters compared to the unitary model, scaling as $N \times D$ , where $N$ is the number of logical qubits (width) and $D$ is the depth of the VMBQC model. This can often make optimization more difficult and may lead to poorly trainable models. In this paper, we present a restricted VMBQC model that extends the unitary setting to a channel-based one using only a single additional trainable parameter. We show, both numerically and algebraically, that this minimal extension is sufficient to generate probability distributions that cannot be learned by the corresponding unitary model.

Figures & tables

Figure 1: Quantum circuit Born machines with VMBQC: (a) Target distribution $Y(x)$ (usually unknown), from which samples are given, $x\sim Y$ . In this paper, the target distribution is generated from the VMBQC model itself with a randomly chosen set of parameters $\{\bm{\theta},\bm{p}\}$ on the cluster state with some given width ( $N$ ) and depth ( $D$ ). (b) The learning model, with width $N$ and depth $D$ , is parametrized by measurement angles $\bm{\theta}:=\{\theta^{j}_{i}\}_{i,j}$ and a single correction probability $p$ , that is, $\bm{p}:=\{p^{j}_{i}=p\}_{i,j}$ , for all partially adapted qubits ( blue ), by which we mean qubits, on which the measurement outcomes are adapted with probability $p$ . When measured on the last layer, the model produces bit strings from the underlying distribution ( $x\sim q_{(\bm{\theta},\bm{p})}$ ) according to the Born rule. The MMD loss takes samples from both the learning model and the target distribution and measures the distance between the two distributions $Y$ and $q_{(\bm{\theta},\bm{p})}$ , and updates the parameters using the parameter shift rule, on a classical computer. The model receives the updated parameters, and this loop continues until the MMD converges to some minimum value. Once converged, the optimal parameters $\bm{\theta}^{*}$ and $\bm{p}^{*}$ generate the optimal learned distribution. The entire learning process can also be implemented using a different learning model (c) where we allow partially uncorrected qubits ( $p^{j}_{i}<1$ ) only on the first layer of the cluster state ( blue ), with the same $p^{j}_{i}=p<1$ for all of them, and the training procedure remains identical. All other qubits, in white , have $p^{j}_{i}=1$ , i.e., measurements on those qubits are always corrected. All the models in this study have periodic boundary conditions, i.e., the first and the last qubits in the cluster state are connected. The qubits in green at the end of the cluster are the output qubits measured in the computational basis.

Figure 2: C-QCA based circuit translation of an MBQC on cluster state: An MBQC on the cluster state (top), with width

N=5

and depth

D=3

, can be understood in the circuit picture (bottom) where

XY

-plane measurements become

Z

-rotations in the circuit, interleaved with Clifford blocks

T_{c}

. The fourth column (top), with output qubits ( green ), is measured in the computational

Z

-basis. An unintended measurement outcome, e.g., when the

(i,j)=(3,1)

qubit (blue) in the cluster state is measured in the

e^{i\theta_{3}^{1}Z}\ket{-}\bra{-}e^{-i\theta_{3}^{1}Z}

-basis, yields the outcome

-1

, resulting in an equivalent byproduct operator

Z_{i}^{s_{i}^{j}}

appearing in the circuit. The qubits in white yield a

+1

outcome. All models have periodic boundary conditions, i.e., the first(top) and last(bottom) qubits are connected with a

CZ

gate.

Figure 3: Propagation of byproducts in $\tilde{\mathcal{E}}_{c}(\bm{\theta},\bm{p})$ model: Top : Consider a byproduct $Z_{3}^{s^{1}_{3}}$ on qubit $3$ at position $(3,1)$ . Propagating this through the cluster state via repeated stabilizer relations generates $X_{i}^{s^{1}_{3}}$ -byproducts on qubits $i$ within the forward light-cone ( green ), which flips (if $s^{1}_{3}=1$ ) the measurement angles of the qubits. For instance, the angle $\theta^{2}_{3}$ at position $(3,2)$ is transformed to $(-1)^{s^{1}_{3}}\theta^{2}_{3}$ , following from $X^{s^{1}_{3}}_{3}R_{z}(\theta^{2}_{3})\ket{\pm}\bra{\pm}R_{z}^{\dagger}(\theta^{2}_{3})X^{s^{1}_{3}}_{3}=R_{z}((-1)^{s^{1}_{3}}\theta^{2}_{3})\ket{\pm}\bra{\pm}R_{z}^{\dagger}((-1)^{s^{1}_{3}}\theta^{2}_{3})$ up to a global phase. After propagating through all layers in the cluster, the byproducts accumulate at the final layer (before computational measurements), forming a Pauli string of $X^{s}$ and $Z^{s}$ byproducts which depend on outcomes $\bm{s}$ . Bottom : Equivalent circuit representation of byproduct propagation via repeated Clifford circuits $T_{c}^{D-j+1}$ ( $D=3$ , $j\in[D]$ ) for a byproduct $Z^{s^{j}_{i}}_{i},\penalty\ i\in[N]$ . In both representations, the accumulated byproducts, at the end of the computation, form a Pauli-string, and the additional $P_{D+1}(\bm{\tilde{s}})$ -layer removes this Pauli-string (Clifford part) from the computation and leaves the non-Clifford effect of the byproducts in the intermediate angle flips.

Figure 4: VMBQC models with one correction probability: We consider four variants of the VMBQC model in a restricted setting, where the number of trainable correction probabilities is limited to a single parameter, i.e.,

\bm{p}:=\{p^{j}_{i}=p\}_{i,j}

. (a) Channel model

\mathcal{E}_{c}^{(A)}

corresponding to Eq. ( 8 ) where measurement outcomes of all qubits ( blue ) are partially adapted and all are sharing the same correction probability

p<1

. (b) Channel model

\mathcal{E}_{c}^{(B)}

corresponding to Eq. ( 9 ) where partially corrected qubits are restricted only to the first layer ( blue ), each with the same correction probability

p<1

, while all remaining qubits ( white ), when measured, are always fully corrected (

p=1

). (c) Channel model

\mathcal{E}_{c}^{(C)}

corresponding to Eq. ( 10 ) where exactly one partially adapted qubit appears in each layer ( blue ), repeated across all layers (chosen as the central qubit). (d) Channel model

\mathcal{E}_{c}^{(D)}

corresponding to Eq. ( 11 ) has only a single qubit, depicted in blue , whose measurement outcome is partially adapted with

p<1

. All the models considered above have periodic boundary conditions, and the qubits in green are the output qubits measured in the computational basis.

Figure 5: Learning performances of VMBQC models: Four families of channel models, defined in Eqs. ( 8 )–( 11 ), together with the unitary model in Eq. ( 1 ), are trained to approximate a target distribution generated by the channel $\mathcal{E}_{c}(\bm{\theta}_{t},\bm{p}_{t})$ in Eq. ( 5 ). Top row: The target model has width $N=7$ and depth $D=6$ , and generates target samples $x\sim Y_{(\bm{\theta}_{t},\bm{p}_{t})}(x)$ according to the born rule (in blue box ). The parameters $\bm{\theta}_{t}=\{\theta^{j}_{i}\in[0,2\pi)\}$ and $\bm{p}_{t}=\{p^{j}_{i}\in[0,1)\}$ are drawn independently and uniformly from their specified ranges. Middle row (a-d): Learning performance of the unitary model $U_{c}$ (yellow) compared to the channel models $\mathcal{E}^{(A)}_{c}$ , $\mathcal{E}^{(B)}_{c}$ , $\mathcal{E}^{(C)}_{c}$ , and $\mathcal{E}^{(D)}_{c}$ (blue), all with $N=7$ and $D=5$ . The qubits ( green ) in the last layer $D=6$ ( bottom row ) correspond to the output qubits. Curves show the mean MMD loss over $200$ epochs, averaged across $10$ random initializations; shaded regions indicate standard deviations. Bottom row: Corresponding channel-based training models, each with a single trainable correction probability $p$ , as specified in Eqs. ( 8 )–( 11 ). The depicted values of $p$ in the figure are not the optimal values found in our experiments but randomly chosen for illustration.

Figure 6: Effect of a single byproduct: Here we examine how well the unitary model $U_{c}$ can learn the channel model $\tilde{\mathcal{E}}^{(D)}_{c}(\bm{\theta},\bm{p})$ when only the measurement outcome of a single qubit is probabilistically adapted (see Eq. ( 15 )). The $x$ -axis labels the different target models, while the $y$ -axis reports the distribution of minimum MMD losses achieved by $10$ independently initialized unitary learning models $U_{c}$ over $200$ epochs. (a) Bottom: The target distribution is generated by a unitary model with measurement angles $\bm{\theta}=\{\theta^{j}_{i}\in[0,1)\}_{i,j}$ drawn uniformly at random. A unitary model is then trained to approximate this distribution. Top (left-most): The distribution of minimum MMD losses is shown by the box labeled by $U_{c}$ on the $x$ -axis, with a single outlier indicated in white . (b) Bottom: A single partially uncorrected qubit (in blue with $p^{l}_{q}=0.135$ ) is introduced, while keeping $\bm{\theta}$ unchanged, to generate a target distribution. Top (second from left): The corresponding MMD distribution is shown with the $\tilde{\mathcal{E}}_{c}^{(D)}(l=1)$ labeled box. (c) Bottom: The same rotation angles are used, while the partially uncorrected qubit is moved to $(l,q)=(2,4)$ with the same $p^{l}_{q}$ as in (b) , yielding the MMD distribution shown by the $\tilde{\mathcal{E}}_{c}^{(D)}(l=2)$ labeled box. (d) Bottom: Moving the partially uncorrected qubit to $(l,q)=(3,4)$ while keeping the rest the same as in (a-c) . Top (right-most): The resulting MMD distribution is shown by the $\tilde{\mathcal{E}}_{c}^{(D)}(l=3)$ labeled box, with an outlier in white .

Figure 7: Effect of two byproducts: Here we examine how well the unitary model $U_{c}$ can learn the channel model $\tilde{\mathcal{E}}_{c}(\bm{\theta},\bm{p})$ with two partially uncorrected qubits whose measurement outcomes are probabilistically adapted. The $x$ -axis labels the different target models, while the $y$ -axis reports the distribution of minimum MMD losses achieved by $10$ independent randomly initialized unitary models $U_{c}$ over $200$ epochs. (a) Bottom: The target distribution is generated from the unitary model itself with $\bm{\theta}=\{\theta^{j}_{i}\in[0,1)\}_{i,j}$ drawn uniformly randomly. Top (left-most): The resulting minimum MMD distribution is shown with the box labeled by $U_{c}$ on the $x$ -axis. (b) Bottom: We choose two uncorrected qubits (in blue ) for some fixed $p^{1}_{3}=p^{1}_{5}=0.1$ for both of them, keeping the $\theta$ same as earlier. Top (second from left): The minimum MMD losses, for the corresponding output distribution, are shown using the $\tilde{\mathcal{E}}_{c}(l=1)$ labeled box. (c) Bottom: The uncorrected qubits are moved to $\penalty\ l=2,\penalty\ q=3,5$ , with the same $p^{l}_{q}$ and $\bm{\theta}$ as before. Top (Third from left): The corresponding MMD distribution, for this target model, is shown by the $\tilde{\mathcal{E}}_{c}(l=2)$ labeled box. (d) Bottom: The uncorrected qubits are placed in layer $l=3$ and the resulting MMD distribution is shown with the corresponding $\tilde{\mathcal{E}}_{c}(l=3)$ labeled box.

Quantum Generative Modeling

Quantum generative modeling uses quantum circuits and quantum-inspired methods to learn and sample data distributions and quantum states.

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Near-term quantum hardware limits circuit depth and often imposes geometrically local connectivity for quantum generative models, restricting the output distributions accessible to shallow unitary Born models. Introducing stochasticity into a unitary quantum Born model can improve the empirical generative performance of the resulting channel model and, for a restricted small-scale architecture, has been proven to represent a strictly larger family of distributions than its unitary counterpart. However, whether such randomness provides a provable separation at fixed shallow depth for arbitrarily large systems has remained open. Here, we show that shared classical randomness, a comparatively weak resource from entanglement theory, is sufficient to establish such a strict scalable representational separation over the corresponding shallow unitary Born model. More specifically, we augment bounded-connectivity shallow unitary circuits, followed by computational-basis measurements, with spatially separated local Pauli operations, whose joint application is controlled by a single classically sampled random bit. The resulting shallow-depth channel model generates long-range correlations in the classical output distribution that no purely unitary shallow-depth model with bounded connectivity can reproduce. For one-dimensional nearest-neighbour architectures, reproducing such distributions with a purely unitary model can require depth

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Arunava Majumder, Marius Krumm, Hendrik Poulsen Nautrup +1

University of Innsbruck, Department of Theoretical Physics, Technikerstraße 21a, A-6020 Innsbruck, Austria.

Quantum Generative Modeling Quantum Circuit Expressibility Quantum Machine Learning Generative Modeling

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Apr 1, 2026

Learning and Generating Mixed States Prepared by Shallow Channel Circuits

Learning quantum states from measurement data is a central problem in quantum information and computational complexity. In this work, we study the problem of learning to generate mixed states on a finite-dimensional lattice. Motivated by recent developments in mixed state phases of matter, we focus on arbitrary states in the trivial phase. A state belongs to the trivial phase if there exists a shallow preparation channel circuit under which local reversibility is preserved throughout the preparation. We prove that any mixed state in this class can be efficiently learned from measurement access alone. Specifically, given copies of an unknown trivial phase mixed state, our algorithm outputs a shallow local channel circuit that approximately generates this state in trace distance. The sample complexity and runtime are polynomial (or quasi-polynomial) in the number of qubits, assuming constant (or polylogarithmic) circuit depth and gate locality. Importantly, the learner is not given the original preparation circuit and relies only on its existence. Our results provide a structural foundation for quantum generative models based on shallow channel circuits. In the classical limit, our framework also inspires an efficient algorithm for classical diffusion models using only a polynomial overhead of training and generation.

Fangjun Hu, Christian Kokail, Milan Kornjača +5

QuEra Computing Inc. · School of Engineering and Applied Sciences, Harvard University · JILA and Department of Physics, CU Boulder

Quantum Generative Modeling Quantum State Preparation Quantum Machine Learning Quantum Many-Body Physics

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Aug 11, 2026

Generative Learning for Quantum Measurement Design

Extracting quantum information from a quantum state is a fundamental task of quantum computation, often requiring the estimation of many non-commuting observables under a finite measurement budget. For both near-term and early fault-tolerant settings, the measurement protocol must balance statistical efficiency against implementation resources such as circuit depth, connectivity, and entangling-gate count. Many existing strategies focus on two extremes: hardware-friendly product measurements with high sampling cost, and fully commuting measurements with deep circuits. Here we recast resource-constrained measurement design as a generative learning problem. We introduce FlowMeas, which uses a generative flow network to directly sample finite ensembles of shallow Clifford measurement circuits subject to a prescribed shot budget and hardware constraints. At zero entangling depth, FlowMeas learns qubit-wise commuting measurement schedules and already matches or improves leading product-measurement methods on nearly all molecular benchmarks. Allowing one or two entangling gate layers yields further reductions in energy estimation error of up to

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Mila – Québec AI Institute, Montréal, QC, Canada · Département d’informatique et de recherche opérationnelle, Université de Montréal, Montréal, QC, Canada · Institut quantique, Université de Sherbrooke, Sherbrooke, QC, Canada +3

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Abstract

Quantum Generative Modeling

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