Optimal stochastic modeling with unitary quantum dynamics

Isolating past information relevant for future prediction is central to quantitative science. Quantum models offer a promising approach, enabling statistically faithful modeling while using less past information than any classical counterpart. Here we introduce a class of phase-enhanced quantum models, representing the most general means of simulating a stochastic process unitarily in causal order. The resulting constructions surpass previous state-of-art methods—both in reducing the information they need to store about the past and in the minimal memory dimension they require to store this information. Moreover, these two features are generally competing factors in optimization—leading to an ambiguity in optimal modeling that is unique to the quantum regime. Our results simultaneously offer quantum advantages for stochastic simulation and illustrate further qualitative differences between classical and quantum notions of complexity.

Figure 1

Unitary quantum model produces a statistical sequence of outputs $\vec{x}=x_0{x}_1{x}_2...$ by interacting a blank ancilla with the memory through $U$ at each time step, and then measuring the state of the output branch in the computational basis.

Figure 2

General three-state Markov model: the notation $y|T_{yx}$ indicates that the transition from state $x$ to $y$ occurs with probability $T_{yx}$ and the output symbol is $y$.

Figure 3

(a) Symmetric three-state quasicycle with slippage $\delta$. (b) $C_q$ and ${C_q^{\phi }}_{min}$ as a function of $p$ for $\delta =0$. (c) The dependence of $C_q^{\phi }$ on the phase factors when $p=0.3$ with $\delta =0$.

Figure 4

In (a) we show the regions with entropic (color plot) and dimensional (dashed line) advantages. The yellow region delineates the nonphysical parameter regime. In (b) we plot $C_q^{\phi }$ in the parameter region denoted by the red segment of the dashed line in (a) and show that the choice of phases that lead to ${C_q^{\phi }}_{min}$ does not always correspond to ${D_q^{\phi }}_{min}$, giving rise to an ambiguity of optimality.

Figure 5

Transition probabilities out of each state in a three-state Markov process can be represented as a point on the positive octant of a unit sphere. Three such points define the process. We sweep over a discretized set of such points in our numerical treatment to systematically study these processes.