Research ArticleQuantum Mechanics

# Experimentally modeling stochastic processes with less memory by the use of a quantum processor

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Science Advances  03 Feb 2017:
Vol. 3, no. 2, e1601302
DOI: 10.1126/sciadv.1601302

### Figures

• Fig. 1 Representation of a stochastic system, with classical and quantum statistical models, at the (j + 1)th time step of evolution.(A) The example system is a pair of switches whose settings determine the value of an output bit and are randomized by a probabilistic process during the step (see text for details). (B) Because the output is determined solely by the parity of the switches, a 1-bit classical model can be used to represent the system and to produce equivalent output statistics. In the example shown, the orientation (up or down) of the vector [or equivalently its color (red or blue)] represents the state of the model and determines the output bit xj + 1. (C) A quantum model allows for reduced complexity (see text for details) by encoding the state into nonorthogonal quantum states (the multicolor vectors, with nonpolar orientations, represent quantum superpositions of logical states). (D) A conceptual classical circuit (double lines represent classical bit rails) for realizing the operation of the classical model above. The classical input state (top rail) at time tj is subjected to a probabilistic action, potentially flipping the state. The model state is then correlated with a meter bit (bottom rail), initially in the logical zero state, via a controlled-NOT (CNOT) gate. Reading out the meter via a logical (Pauli “Z”) measurement provides the output bit. After readout, the model state is passed on to the next time step, along with a fresh meter bit. (E) A conceptual quantum circuit (glowing lines represent qubit states) for realizing the quantum model above. The operation is similar to the classical circuit in (D), except that the probabilistic action is delayed until the readout of the meter (as above), which yields a random result, and collapses the model state because of the entanglement generated by the CNOT gate acting on superposition states. (F and G) Conceptual circuits of the classical and quantum models, as experimentally realized. The key difference (for practical reasons only) is the interruption of the model states for characterizing measurements (denoted T, for quantum state tomography) with subsequent repreparation.
• Fig. 2 Replicating statistical behavior with causal states—transition diagram for the model.(A) In the example in Fig. 1, the probability of the model state bit transitioning from one causal state (denoted here by a circle) to the other is P, and thus, the probability of remaining in the same state is 1 − P. (B) In general, a two–causal state model may have a transition probability, either P→ or P←, that depends on the causal state at the beginning of the step. The case we consider for most of this work, P = P→ = P←, is a particular example.
• Fig. 3 Experimental setup.Photons from a continuous wave–pumped spontaneous parametric downconversion (SPDC) source are prepared in the relevant input states by half-wave plates (HWPs) and are incident on a linear optics CNOT gate realized with partially polarizing beam splitters (PPBS) (see Materials and Methods). Converting between classical and quantum models requires changing the input states from classical (orthogonal) polarization states to nonorthogonal superposition states. Measurement of one output determines the output bit at the current time step, and the other output is tomographically characterized over many measurement runs to determine the state of the model and its entropy. Key elements include polarizing beam splitters (PBS), PPBS, quarter wave plates (QWP) and HWP, and avalanche photodiode (APD) single-photon detectors.
• Fig. 4 Experimental data for classical and quantum models of the stochastic process.(A) Experimentally measured statistical complexities (entropy) for the classical and quantum model states, sampled for a range of values for P(=P→ = P←). Blue squares are the quantum data, and black diamonds are the classical data. The orange solid line represents the theoretically calculated entropy for the classical scenario. The black curve represents the theoretically calculated entropy for the quantum scenario. Error bars are 1 SD, derived from Poissonian counting statistics. Error bars not shown are much smaller than the data points. The orange triangle denotes the classical prediction for P = 0.5, where no memory is required for the corresponding completely random process. (B) Real parts of the tomographically determined equilibrium density matrix, at the model state output, for P = 0.8. Top, classical model; bottom, quantum model; left, theory; right, experiment. Imaginary components (small) are not shown. Fractional uncertainties in the density matrix reconstruction are comparable in size to the uncertainties in (A).