Exact block encoding of imaginary time evolution with universal quantum neural networks

TL;DR

The paper introduces exact, block-encoded quantum neural network approaches (RBM and DBM variants) to represent the imaginary-time evolution $e^{- au H}$ for arbitrary qubit Hamiltonians. By employing auxiliary-field decompositions and unitary Boltzmann-machine architectures, it provides analytic parameter expressions that avoid stochastic optimization and proves universality under a universal gate set, with linear scaling of resources in system size and total imaginary time. The authors demonstrate the methods numerically on a small transverse Ising model, analyze post-selection success probabilities, and discuss hardware implementations via mid-circuit measurements. These results offer a non-variational, scalable route to thermal states and ground-state preparation on quantum devices, potentially mitigating the sign problem in classical simulations through quantum hardware execution.

Abstract

We develop a constructive approach to generate quantum neural networks capable of representing the exact thermal states of all many-body qubit Hamiltonians. The Trotter expansion of the imaginary-time propagator is implemented through an exact block encoding by means of a unitary, restricted Boltzmann machine architecture. Marginalization over the hidden-layer neurons (auxiliary qubits) creates the non-unitary action on the visible layer. Then, we introduce a unitary deep Boltzmann machine architecture, in which the hidden-layer qubits are allowed to couple laterally to other hidden qubits. We prove that this wave function ansatz is closed under the action of the imaginary-time propagator and, more generally, can represent the action of a universal set of quantum gate operations. We provide analytic expressions for the coefficients for both architectures, thus enabling exact network representations of thermal states without stochastic optimization of the network parameters. In the limit of large imaginary time, the ansatz yields the ground state of the system. The number of qubits grows linearly with the system size and total imaginary time for a fixed interaction order. Both networks can be readily implemented on quantum hardware via mid-circuit measurements of auxiliary qubits. If only one auxiliary qubit is measured and reset, the circuit depth scales linearly with imaginary time and system size, while the width is constant. Alternatively, one can employ a number of auxiliary qubits linearly proportional to the system size, and circuit depth grows linearly with imaginary time only.

Figures (5)

• Figure 1: Graphical representations of the RBM identities for (a) one-body, (b) two-body, and (c) three-body interactions. Open circles denote non-unitary one-body interactions. Double lines represent non-unitary interactions between two or more qubits. Single lines denote unitary interactions between visible and hidden qubits. In sub-figure (c), the inset boxes show the induced one- and two-body couplings between the three visible qubits. Within each inset, the ordering of the RBM indices is the same as above, although the labels have been omitted for simplicity.
• Figure 2: Quantum circuit diagram that implements the two-body identity in Eq. \ref{['eq:2body_id']} using block encoding.
• Figure 3: Two representations of the same physical system of 3 visible qubits $\sigma_i^z$ expressed in (a) DBM ansatz of Eq. (\ref{['eq:DBM_ansatz']}) and (b) L-DBM ansatz of Eq. (\ref{['eq:DBM_lat_ansatz']}). Lines between visible (hidden) and hidden (deep) units represent interlayer couplings $W_{ij}$ ($W'_{ij}$). Lines between hidden units correspond to lateral couplings $L_{ij}$. Dashed lines denote encodings that are equivalent after marginalization over hidden unit $d_2$. Dotted lines indicate that deep unit $d_1$ in the DBM is relabeled as laterally coupled hidden unit $h_0$ in the L-DBM architecture. Additional one-body bias terms ($a_i,b_i,b_i'$) are not represented explicitly.
• Figure 4: Illustration of the action of the rotation operators $(H, H^y, {H^y}^{\dag})$ on the L-DBM ansatz. (i) Rotation operator is applied to visible qubit $l$. (ii) Add one hidden qubit $h_{M+1}$ that couples to visible qubit $l$ and couples laterally to all hidden units currently connected to visible qubit $l$. (iii) Remove all couplings between visible qubit $l$ and existing hidden layer qubits. (iv) Adjust the one-body coupling of visible qubit $l$. The exact values of the required couplings differ depending on the operator under consideration (see text).
• Figure 5: One dimensional transverse Ising model at the critical point with three qubits and periodic boundary conditions at finite temperature. (a) Energy expectation value as function of imaginary time (inverse temperature) and (b) success probability for post-selection.

Theorems & Definitions (15)

• Theorem 1
• proof
• Corollary 1.1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Lemma 3.1
• Lemma 3.2