Thermodynamic coprocessor for linear operations with input-size-independent calculation time based on open quantum system

TL;DR

A direct mapping between open quantum systems and electrical crossbar structures frequently used in analog vector-matrix multiplication is constructed, showing that dissipation rates multiplied by open quantum system's modes frequencies can be seen as conductivities, reservoirs'occupancies can be seen as potentials, and stationary energy flows can be seen as electric currents.

Abstract

Linear operations, e.g., vector-matrix and vector-vector multiplications, are core operations of modern neural networks. To diminish computational time, these operations are implemented by parallel computations using different coprocessors. In this work we show that an open quantum system consisting of bosonic modes and interacting with bosonic reservoirs can be used as an analog {thermodynamic} coprocessor implementing multiple vector-matrix multiplications with stochastic matrices in parallel. Input vectors are encoded in occupancies of reservoirs, and the output result is presented by stationary energy flows. The operation takes time needed for the system's transition to a non-equilibrium stationary state independently on the number of the reservoirs, i.e., on the input vector dimension. With technological limitations being considered, a device of $5\times5$ cm$^2$ area covered with the coprocessors can conduct of the order of $10^{11}$ operations per second per a mode of the OQS. The computations are accompanied by an entropy growth. We construct a direct mapping between open quantum systems and electrical crossbar structures frequently used in analog vector-matrix multiplication, showing that dissipation rates multiplied by open quantum system's modes frequencies can be seen as conductivities, reservoirs' occupancies can be seen as potentials, and stationary energy flows can be seen as electric currents.

Figures (4)

• Figure 1: Schematic representation of the considering OQS.
• Figure 2: Electrical circuit supporting currents equivalent to the energy flow ${J}_{\kappa,j}(\vec{T})$ in Eq. (\ref{['J_el']}) for a particular $\kappa$, i.e., the energy flows through a frequency of the OQS.
• Figure 3: Electrical circuit supporting currents equivalent to the energy flows ${J}_{\kappa,j}(\vec{T})$ in Eq. (\ref{['J_el']}). This circuit is the circuit from Fig. \ref{['Scheme']} repeated for each mode of the OQS.
• Figure 4: Equivalent CS of the OQS. Potentials $\varphi_{\kappa,j}$ from Fig. \ref{['Scheme1']} are formed by the connection of the wires with the same $j$ by common bars through resistances $r_{\kappa,j}$. The potentials $\Phi_j$ are applied to the bars. The potentials $\Phi_j$ and the resistances $r_{\kappa,j}$ are defined by Eqs. (\ref{['PHI']}) and provide the same currents through the wires as in Fig. \ref{['Scheme1']}.