  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.
