Source-linked AI summary
Constructing exact representations of quantum many-body systems with deep neural networks
Giuseppe Carleo, Yusuke Nomura, Masatoshi Imada
TL;DR
The paper addresses the challenge of obtaining compact exact classical representations of quantum many-body ground states without stochastic parameter optimization. It constructs deep Boltzmann machines whose network updates reproduce imaginary-time evolution exactly, yielding deterministic representations for interacting spin models. The resulting DBM schemes also provide an alternative to path-integral representations and can improve variational RBM states within the stated scope.
Problem
Existing neural-network approaches commonly use shallow RBMs with numerically optimized parameters, motivating exact and efficient classical representations of quantum many-body states.
Method
The paper uses a two-layer deep Boltzmann machine and deterministic local network updates derived from imaginary-time propagators to represent evolving quantum states.
Results
The construction exactly represents ground states of interacting spin models, including transverse-field Ising and Heisenberg models, without further variational optimization.
Takeaways & Limitations
DBM representations offer an alternative to path-integral formulations and can systematically improve existing RBM variational results.
Takeaways & Limitations
The explicit construction includes assumptions on the wave-function structure and parameter choices in the Heisenberg constructions.
Abstract
from arXiv · showhide
We develop a constructive approach to generate artificial neural networks representing the exact ground states of a large class of many-body lattice Hamiltonians. It is based on the deep Boltzmann machine architecture, in which two layers of hidden neurons mediate quantum correlations among physical degrees of freedom in the visible layer. The approach reproduces the exact imaginary-time Hamiltonian evolution, and is completely deterministic. In turn, compact and exact network representations for the ground states are obtained without stochastic optimization of the network parameters. The number of neurons grows linearly with the system size and total imaginary time, respectively. Physical quantities can be measured by sampling configurations of both physical and neuron degrees of freedom. We provide specific examples for the transverse-field Ising and Heisenberg models by implementing efficient sampling. As a compact, classical representation for many-body quantum systems, our approach is an alternative to the standard path integral, and it is potentially useful also to systematically improve on numerical approaches based on the restricted Boltzmann machine architecture.
INTRODUCTION
The paper introduces a deterministic construction of exact deep Boltzmann machine representations for quantum ground states by tracking imaginary-time evolution through local network updates. Applied to transverse-field Ising systems, the procedure adds or modifies hidden and deep units to reproduce propagators exactly.
- Alternative classical representations of quantum states support conceptual and numerical approaches to challenging many-body problems.
- The approach constructs exact deep network structures for interacting lattice-spin ground states without variational optimization of network parameters.
- Imaginary-time evolution is decomposed into short-time propagators, whose effects are represented by dynamically modifying DBM parameters and structure.
- The DBM uses physical spins, hidden units, and deep units arranged in two hidden layers, with no same-layer connections.
- Transverse-Field Ising model: For the transverse-field Ising model, the diagonal propagator adds a hidden unit mediating a physical-spin interaction, while the off-diagonal propagator adds a deep unit and hidden unit.
- Transverse-Field Ising model: Repeating the exact updates for the diagonal and off-diagonal parts implements the full imaginary-time evolution.
1 deep, 3 hidden
The 1d-3h construction represents each Heisenberg bond propagator by adding one deep neuron and three hidden neurons, including an imaginary-coupling constraint step.
- The 1d-3h construction adds one deep neuron and three hidden neurons to satisfy the Heisenberg bond-propagator equation.
- The construction proceeds by connecting the new deep unit to existing hidden units, rewiring physical-spin connections, and introducing two interaction-mediating hidden units.
- A final hidden unit uses imaginary-valued interactions to enforce the deep-spin constraint through the iπ/6 trick.
2 deep, 6 hidden
The 2d-6h construction introduces two deep units and six hidden units per Heisenberg bond, using auxiliary-spin constraints that connect the DBM evolution to a path-integral representation.
- The 2d-6h construction introduces two auxiliary deep spins per bond with the constraint d[l] + d[m] = σz_l + σz_m.
- The construction reconnects existing hidden units, adds four mediating hidden units, and then adds two constraint-enforcing hidden units with imaginary-valued weights.
- After tracing out hidden neurons, the representation is equivalent to path-integral Monte Carlo, with deep neurons acting as additional classical spins at each time slice.
- Figure 3 compares three explicit Heisenberg constructions: 1d-3h, 2d-6h, and 2d-4h.
2 deep, 4 hidden
The DBM construction uses auxiliary deep and hidden neurons to implement imaginary-time evolution while preserving a two-layer network structure. These representations support sampling-based evaluation of physical quantities and reproduce the target evolution in spin-model examples.
- 2 deep, 4 hidden: The 2d-4h construction introduces two deep variables and four hidden units around each bond propagator.The construction adds d[l], d[lm], h[l], h[m], h[lm1], and h[lm2].
- 2 deep, 4 hidden: Imaginary-time evolution updates existing couplings and introduces new couplings between deep, hidden, and physical variables.The update includes revised W couplings and additional interactions involving the newly introduced neurons.
- 2 deep, 4 hidden: The construction enforces the constraint d[l]d[lm] = σz_lσz_m through a DBM-form representation involving hidden neurons.The constraint is represented using the hidden variables h[lm1] and h[lm2].
- 2 deep, 4 hidden: The three constructions differ in topology: 2d-6h is local, 1d-3h is partly non-local, and 2d-4h is non-local in both layers.These topology differences distinguish how each construction distributes interactions across the two hidden layers.
- Physical-quantity extraction: Physical observables are evaluated by decomposing DBM weights and sampling a pseudo-probability distribution with Gibbs, Metropolis, or cluster updates.The sampling distribution includes physical, hidden, and deep variables and their primed counterparts.
- Numerical results: For the one-dimensional TFI and Heisenberg models, sampled DBM energies closely follow exact imaginary-time evolution, while a pre-optimized RBM reduces the time needed to reach the Heisenberg ground state.The TFI check uses an empty network, and the Heisenberg check uses the 2d-6h construction.
- Discussion: Exact ground states can be constructed with two hidden layers without further variational optimization, and DBM schemes can improve existing RBM results.For bipartite Heisenberg lattices, the explicit algorithms yield sign-problem-free representations.
1d-3h construction
The 1d-3h construction updates the DBM locally around a bond while using newly introduced neurons to mediate the required interactions and constraints.
- 1d-3h construction: The updated wave function for the TFI construction is defined through a solution of the bond-evolution condition.The construction specifies how the network changes after applying the bond propagator.
- 1d-3h construction: Connections from one spin can be cut and reassigned to the hidden units connected to the other spin, with the corresponding interaction updated.This rewiring implements the first condition of the construction.
- 1d-3h construction: Additional hidden neurons transform direct spin-deep interactions into the standard DBM form.The construction introduces h[lm1] and h[lm2] using the two-body gadget.
- 1d-3h construction: A third hidden neuron with complex couplings enforces the required constraint.The added neuron is h[lm3], with pure complex iπ/6 couplings.
2d-6h construction
The 2d-6h construction represents bond evolution by introducing six hidden neurons and corresponding deep-neuron interactions, with the resulting terms mediated within the DBM architecture.
- 2d-6h construction: The updated wave function is constructed by solving the bond-evolution equation for the 2d-6h network.The construction is specified through the form of the new wave function and a solution of the governing equation.
- 2d-6h construction: The relevant interactions are mediated by newly introduced hidden neurons associated with the bond.The construction assigns distinct hidden neurons to the interaction terms generated by the propagator.
- 2d-6h construction: Additional hidden neurons and complex couplings complete the DBM representation of the bond evolution.The construction introduces h[lm5] and h[lm6] to realize the required constraint.
2d-4h construction
The 2d-4h construction uses auxiliary deep and hidden variables to encode bond evolution, then restores the standard DBM form through coupling transformations and gauge choices.
- 2d-4h construction: The construction assumes a specific post-propagator wave-function structure and seeks parameters solving the associated evolution equation.The derivation introduces solutions for updated couplings and auxiliary variables.
- 2d-4h construction: The updated physical-hidden couplings are expressed through increments to the original couplings.The quantities W_nj, Z_lmj, and related parameters determine the transformed network.
- 2d-4h construction: On a bipartite lattice, local gauge choices keep the relevant coupling real and avoid the negative-sign or complex-phase problem.The choices differ between ferromagnetic and antiferromagnetic interactions and sublattices.
- 2d-4h construction: The coupling solutions use inverse hyperbolic functions and exponential expressions to determine transformed network parameters.These expressions provide the parameters needed to represent the evolved state in DBM form.
- 2d-4h construction: The resulting network evolution is described by adding two deep and four hidden variables to the existing neuron set.The notation {h̄, d̄} denotes the combined set of existing and newly introduced neurons.
13 Supplementary Information
The supplementary construction represents imaginary-time evolution with a deep Boltzmann machine by decomposing propagators into exact interaction gadgets and updating network parameters or adding neurons. These rules apply to transverse-field Ising evolution and preserve exactness under repeated short-time steps.
- Interaction gadgets: Interaction gadgets decompose two-, three-, and four-body terms into DBM-compatible couplings among physical, hidden, and deep variables.The four-body identity can generate the three-body and alternative two-body gadgets by fixing selected variables.
- Ground-state construction: Imaginary-time evolution is constructed by applying a second-order Trotter-Suzuki decomposition to short-time propagators for two non-commuting Hamiltonian parts.Each short-time propagator is represented exactly by modifying existing parameters and, when necessary, enlarging the network.
- Interaction propagator: Diagonal interaction propagators are implemented by adding a hidden neuron connected to the relevant visible spins and choosing couplings through the two-body gadget.The resulting parameters satisfy the propagator equation for all visible-spin configurations up to a finite normalization constant.
- Transverse-field Ising model: For transverse-field terms, the construction adds deep and hidden neurons, adjusts couplings, and can reconnect a spin after eliminating its previous hidden interactions.The positive- and negative-field cases are handled separately, with a gauge transformation enabling real interaction weights for negative fields.
B. Heisenberg Model
The Heisenberg construction decomposes the Hamiltonian into bond propagators and represents each bond evolution exactly with several DBM architectures. A gauge transformation on bipartite lattices supports real couplings and removes the negative sign problem for the presented constructions.
- Propagator decomposition: The Heisenberg imaginary-time propagator is decomposed into bond terms using a Trotter-Suzuki procedure.Groups of commuting bonds can be applied exactly, such as odd and even bonds on a one-dimensional chain or checkerboard groups on a square lattice.
- Scope and sign problem: The Heisenberg construction is formulated for general bond couplings and can be extended straightforwardly to anisotropic or disordered bonds.The derivation also discusses zero-bias solutions, gauge transformations, and complex biases as alternative ways to avoid sign problems.
- Exact bond constructions: Three exact DBM representations are constructed for each Heisenberg bond propagator, and repeated sufficiently small steps reach the ground state with controlled accuracy.The constructions represent e−δτ H_lm exactly and can be iterated many times.
- Gauge transformation: For antiferromagnetic couplings on bipartite lattices, a local π rotation changes the signs of σx interactions and enables mostly real DBM couplings.The remaining exceptions are couplings needed to enforce local constraints on deep-neuron spins.
1. 1 deep, 3 hidden (1d-3h) representation
The 1d-3h construction exactly implements Heisenberg bond imaginary-time evolution by adding one deep neuron and three hidden neurons while enforcing a bond-dependent constraint. Its local physical-hidden couplings remain local, whereas deep-layer couplings become nonlocal as evolution proceeds.
- Construction: The deep neuron is constrained by the bond spins, d[lm] = σz_l + σz_m.The constraint is implemented using the added hidden neuron h[lm3] and complex couplings.
- Construction: One deep neuron d[lm] and three hidden neurons are introduced to represent each Heisenberg bond propagator exactly.The construction mediates otherwise forbidden direct interactions and enforces the required bond constraint through additional hidden units.
- Imaginary-time update: Each bond-propagator step cuts spin l's existing hidden connections, reconnects it to hidden units attached to spin m, and inserts new mediated couplings.This update follows from matching hidden-unit interactions before and after the propagator.
- Sampling: The evolved weight remains non-negative for every spin and deep-neuron configuration, so Metropolis sampling avoids negative-sign problems.The proof establishes non-negativity after one bond update and therefore throughout the imaginary-time evolution.
- Coupling structure: As imaginary time advances, physical-hidden W couplings stay local while W′ couplings become nonlocal or long ranged.Figure 5 illustrates the emergence of complex couplings associated with the newly added hidden neuron at each bond evolution.
2. 2 deep, 6 hidden (2d-6h) representation
The 2d-6h construction represents Heisenberg imaginary-time evolution by adding two deep and six hidden neurons per bond update, with complex couplings enforcing the required constraints. Rearranged as imaginary-time layers, the DBM becomes equivalent to the path-integral formulation.
- Construction: The added hidden neurons mediate direct interactions between physical spins, deep neurons, and successive deep layers that are forbidden in the standard DBM architecture.The construction recovers standard DBM connectivity through hidden-neuron gadgets.
- Constraint enforcement: The iπ/4 and iπ/8 trick filters configurations that violate the constraint σz_l + σz_m = d[l] + d[m].Two additional hidden neurons implement this filtering through complex connections.
- Imaginary-time update: At each imaginary-time step, all existing physical-hidden connections vanish and the physical spins reconnect to the newly introduced hidden neurons.The corresponding deep neurons connect to the same new hidden-neuron set.
- Path-integral correspondence: The deep neurons act as classical spin degrees of freedom along imaginary-time layers, making the 2d-6h DBM equivalent to the path-integral formulation.Two deep neurons are introduced per bond at each evolution, and the extended network forms a D + 1 dimensional classical spin system.
3. 2 deep, 4 hidden (2d-4h) representation
The 2d-4h construction extends the DBM with additional deep and hidden variables so imaginary-time bond evolution can be represented in standard DBM form. Repeated updates produce a nonlocal network from which the ground-state wave function can be sampled.
- 2d-4h construction: Complex physical biases can implement a gauge transformation that removes the sign problem for the relevant bipartite-lattice construction.The site-dependent bias remains fixed during imaginary-time evolution and can represent the sublattice gauge choice.
- 2d-4h construction: The strategy adds deep and hidden variables to encode imaginary-time evolution while recovering a standard DBM architecture.Two deep and four hidden variables are introduced for each bond evolution, with supplementary variables enforcing the required transformations.
- 2d-4h construction: The construction updates couplings deterministically for each imaginary-time bond evolution, with separate rules for spin configurations on parallel and antiparallel bonds.The parameters are incremented through explicit coupling transformations, including cases distinguished by σ^z_lσ^z_m.
- 2d-4h construction: After repeated operations, the network becomes nonlocal and contains physical, hidden, and deep variables connected across the lattice.The resulting standard DBM has no direct physical–deep interactions; both couple through hidden variables.
- Sampling: The ground-state wave function is obtained after sufficiently long imaginary-time evolution by analytically summing hidden variables and Monte Carlo sampling deep variables.The sampling procedure targets the generated DBM representation rather than requiring direct summation over all configurations.
- Sampling: Gibbs sampling alternates conditional updates of hidden, visible, and deep variables, with visible-spin updates parallelizable across sites.The overall scheme cycles between hidden-variable sampling and joint visible/deep-variable sampling.
2. Phase problem in the Metropolis scheme
The phase problem depends on the lattice geometry and the treatment of hidden variables. Analytical marginalization can avoid negative weights in some Heisenberg constructions, whereas non-bipartite lattices may retain sampling difficulties.
- Phase problem: Analytically summing hidden variables can eliminate negative weights even when the DBM parameters are complex.This applies to the presented Heisenberg constructions, where complex couplings enforce constraints.
- Phase problem: On non-bipartite lattices, exact DBM imaginary-time solutions remain available, but imaginary couplings can cause a sampling sign problem.A good initial state may shorten the evolution before severe sign problems develop.
- Phase problem: For the Heisenberg model, tracing out hidden spins imposes constraints that can make cluster updates more efficient than local updates.The 2d-6h representation is described as equivalent to the path-integral formalism under the DBM imaginary-time evolution.