arXiv:1602.08159v2

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10 7 6 5 4 3 2 1 Parity check capacity CPC 7 6 5 4 3 2 1 ESN100 ESN50 ESN10 0 0 0 5 10 0 15 5 20 10 25 15 30 20 25 30 Short term memory capacity CSTM (b) x’i (t ) x’i (t ) 1 0.8 0.6 0.4 0.2 0 fully-connected transverse Ising QR (non-integrable system) Parity check capacity CPC 7 7 N=6 7 N=6 N=7 6 (c) N=7 N=6 N=5 6 N=5 N=7 N=4 6 N=4 N=5 N=3 5 N=3 N=4 N=2 5 N=2 N=3 5 N=2 4 ESN500 4 ESN300 4 ESN200 3 ESN100 3 3 SR~1 2 2 ESN50 2 1 1 ESN10 1 0 0 0 5 10 15 20 25 30 0 0 5 10 15 20 25 30 0 5 10 15 20 25 Short term memory capacity CSTM 30 4000 4050 4100 4150 4200 1D transverse Ising QR (integrable system) time tΔ 1 0.8 0.6 0.4 0.2 0 (a) ESN200 N=6 N=7 N=5 N=4 N=3 N=2 ESN300 10 ESN500 7 6 5 5 4 0 128 τΔ SR~1 3 2 -5 integrable systems 1 -10 0.125 0 0 5 10 15 20 25 30 4000 4050 4100 4150 4200 time tΔ qubit 1 qubit 2 qubit 3 qubit 4 qubit 5 purity input FIG. 7. STM and PC capacities under various settings. (a) Capacities for the 5-qubit QRs plotted with various parameters τ∆ = 0.125–128, J/∆ = 0.05–1.0, h/∆ = 0.5, and V = 1–50. Integrable cases with 1DNN couplings are shown as “integrable systems”. For each setting, the capacities are evaluated as an average on 20 samples, and the standard deviations are shown by the error bars. (b) Typical dynamics with fully connected (upper) and 1DNN couplings (lower) are shown with the signals from each qubit. The input sequence and purity, a measure of quantum coherence, are shown by dotted and solid lines, respectively. (c) QRs with J/∆ = 2h/∆ = 1 and 0.125 ≤ τ∆ ≤ 128 for N = 2–7. For each setting, the capacities are evaluated as an average on 20 samples, and the standard deviations are shown by the error bars. The ESNs with 10–500 nodes are shown as references. In Fig. 7 (c), the STM and PC capacities are plotted for the QRs from N = 2 to N = 7. The 7-qubit QRs, for example, with τ∆ = 2, J/∆ = 2h/∆ = 1, and V = 10– 50, are as powerful as the ESNs of 500 nodes with the spectral radius around 1.0. Note that even if the virtual nodes are included, the total number of nodes NV = 350 is less than 500. B. Robustness against imperfections We here investigate the effect of decoherence (noise) to validate the feasibility of QRC. We consider two types of noise: the first is decoherence, which is introduced by an undesired coupling of QRs with the environment, thereby resulting in a loss of quantum coherence, and the other is a statistical error on the observed signals from QRs. The former is more serious because quantum coherence is, in general, fragile against decoherence, which is the most difficult barrier for realizations of quantum information processing. We employ the dephasing noise as decoherence, which is a simple yet experimentally dominant source of noise. In the numerical simulation, the time evolution is divided into a small discrete interval δt, and qubits are exposed to the single-qubit phase-flip channel with probability (1 − e −2γδt )/2 for each timestep: E(ρ) = 1 + e−2γδt ρ + 1 − e−2γδt ZρZ. (19) 2 2 This corresponds to a Markovian dephasing with a de-
11 tau=1 tau=2 tau=4 tau=8 tau=16 tau=32 tau=64 tau=128 tau=1 tau=2 tau=4 tau=8 tau=16 tau=32 tau=64 tau=128 STM capacity CSTM PC capacity CPC 25 25 25 tau=1 tau=1 25 25 25 tau=1 tau=1 tau=1 tau=2 V=1 tau=2 V=2 tau=2 V=5 tau=2 V=10 tau=2 V=25 V=50 20 tau=4 20 tau=4 20 tau=4 tau=4 tau=8 tau=8 20 20 tau=4 tau=8 tau=8 tau=8 20 tau=16 tau=16 tau=16 tau=16 tau=16 15 tau=32 15 tau=32 15 tau=32 15 tau=32 15 tau=32 tau=64 tau=64 tau=64 tau=64 tau=64 15 tau=128 tau=128 tau=128 tau=128 tau=128 10 10 10 10 10 10 25 tau=1 5 5 5 5 5 5 tau=2 tau=1 25 tau=4 0 0 0 20 tau=2 0 tau=8 0 0 2 3 4 5 6 7 2 3 4 5 6 7 252 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 tau=1 tau=4 N N N 20 tau=16 N N N tau=2 tau=8 tau=32 25 15 tau=1 τΔ=1 tau=4 τΔ=4 tau=16 τΔ=16 tau=64 20 τΔ=64 tau=2 τΔ=2 tau=8 τΔ=8 tau=32 τΔ=32 tau=128 τΔ=128 15 10 tau=4 10 tau=16 20 10 tau=64 10 10 10 10 tau=8 tau=1 tau=32 tau=1 tau=128 tau=1 tau=1 tau=2 V=1 tau=2 V=2 15 tau=2 V=5 tau=2 V=10 tau=2 V=25 V=50 tau=16 tau=64 8 tau=4 10 tau=32 8 tau=4 8 tau=4 8 tau=4 8 tau=4 8 tau=8 tau=128 tau=8 15 tau=8 tau=8 tau=8 5 tau=64 tau=16 tau=16 tau=16 tau=16 tau=16 10 tau=128 6 tau=32 6 tau=32 6 tau=32 6 tau=32 6 tau=32 6 tau=64 tau=64 tau=64 tau=64 5 tau=64 tau=128 tau=128 10 tau=128 tau=128 tau=128 0 4 4 4 5 4 4 2 3 4 54 6 7 0 2 2 5 2 2 2 3 4 5 26 7 0 2 2 3 4 5 6 7 0 0 0 0 0 0 2 3 4 5 6 7 2 03 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 2 3 4 5 6 7 N 2 N 3 4 5 6 7N N N N FIG. 8. Scaling of the STM and PC capacities against the number of the qubits. (Top) The STM capacity C STM is plotted against the number of qubits N for each number of virtual nodes (V = 1, 2, 5, 10, 25, 50 from left to right). (Bottom) The PC capacity C PC is plotted against the number of qubits N for each number of virtual nodes (V = 1, 2, 5, 10, 25, 50 from left to right). C PC = 2(N − 2) is shown by dotted lines. The error bars show the standard deviations evaluated on 20 samples of the QRs with respect to the random couplings. phasing rate γ and destroys quantum coherence, i.e. offdiagonal elements in the density matrix. Apart from the dephasing in the z-direction, we also investigate the dephasing in x-direction, where the Pauli Z operator is replaced by X. In Fig. 9, the STM and PC capacities (C STM , C PC ) are plotted for τ∆ = 0.5, 1.0, 2.0, and 4.0 (from left to right) with V = 1, 2, 5, 10, 25, and 50 and γ = 10 −1 , 10 −2 , and 10 −3 . The results show that dephasing of the rates 10 −2 − 10 −3 , which is within an experimentally feasible range, does not degrade computational capabilities. A subsequent increase in the dephasing rate causes the STM capacity to become smaller, especially for the case with a shorter time interval τ∆ = 0.5. On the other hand, the PC capacity is improved by increasing the dephasing rate. This behaviour can be understood as follows. The origin of quantum decoherence is the coupling with the untouchable environmental degree of freedom, which is referred to as a “reservoir” in the context of open quantum systems. Thus, decoherence implies an introduction of another dynamics with the degree of freedom in the “reservoir” computing framework. This leads to the decoherence-enhanced improvement of nonlinearity observed in Fig. 9, especially for a shorter τ with less rich dynamics. Of course, for a large decoherence limit, the system becomes classical, preventing us from fully exploiting the potential computational capability of the QRs. This appears in the degradation of the STM capacity. By attaching the environmental degree of freedom, the spatialized temporal information is more likely to leak outside the true nodes. Accordingly we cannot reconstruct a past input sequence from the signals of the true nodes. In other words, quantum coherence is important to retain information of the past input sequence within the addressable degree of freedom. In short, in the QRC approach, we do not need to distinguish between coherent dynamics and decoherence; we can exploit any dynamics on the quantum system as it is, which is monitored only from the addressable degree of freedom of the quantum system. Next, we consider the statistical noise on the observed signal from the QRs. We investigate the STM and PC capacities by introducing Gaussian noise with zero mean and variance σ on the output signals as shown in Fig. 10. The introduction of statistical noise leads to a gradual degradation of the computational capacities. However, the degradation is not abrupt, which means that QRC would be able to function in a practical situation. In the small τ region, the STM capacity is sensitive to the statistical observational noise. This is because in such a region, the dynamic range of the observed signals be-
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