Temporal Information Processing on Noisy Quantum Computers

The combination of machine learning and quantum computing has emerged as a promising approach for addressing previously untenable problems. Reservoir computing is an efficient learning paradigm that utilizes nonlinear dynamical systems for temporal information processing, i.e., processing of input sequences to produce output sequences. Here we propose quantum reservoir computing that harnesses complex dissipative quantum dynamics. Our class of quantum reservoirs is universal, in that any nonlinear fading memory map can be approximated arbitrarily closely and uniformly over all inputs by a quantum reservoir from this class. We describe a subclass of the universal class that is readily implementable using quantum gates native to current noisy gate-model quantum computers. Proof-of-principle experiments on remotely accessed cloud-based superconducting quantum computers demonstrate that small and noisy quantum reservoirs can tackle high-order nonlinear temporal tasks. Our theoretical and experimental results pave the path for attractive temporal processing applications of near-term gate-model quantum computers of increasing fidelity but without quantum error correction, signifying the potential of these devices for wider applications including neural modeling, speech recognition, and natural language processing, going beyond static classification and regression tasks.

Figure 1

Quantum circuit interpretation of the QR universal subclass described in Sec. 4a. Here ${\rho }_{l-1}^{(k)}$ and ${\sigma }_k$ are two quantum registers (i.e., groups of qubits), whereas $\rho (u_l)$ and ${\rho }_{{ϵ}_k}$ are two single-qubit states. The unitaries $U_1^{(k)}$ and $U_0^{{(k)}^{†}}$ act on ${\rho }_{l-1}^{(k)}$, controlled by $\rho (u_l)$. The right-most operation (the $SW$) swaps the states of ${\overline{\rho }}_{l-1}^{(k)}$ and ${\sigma }_k$, controlled by ${\rho }_{{ϵ}_k}$.

Figure 2

Quantum circuit schematics for (a) $U_0(\mathit{\theta })$ and (b) $U_1(\mathit{\varphi })$ employed in proof-of-principle experiments, described by Eq. (4) in Sec. 6a. Here $j_t$ and $j_c$ are the target and control qubits, respectively. The unitaries $U_0(\mathit{\theta }),U_1(\mathit{\varphi })$ consist of $N_0,N_1$ layers of highlighted gate operations, with each layer acting on a different qubit pair ($j_t,j_c$).

Figure 3

Qubit coupling maps of the IBM superconducting quantum processors. (a) The 20-qubit Boeblingen device. (b) Both the 5-qubit Ourense and Vigo devices.

Figure 4

The QR’s predicted outputs for (a) the multistep prediction problem and (b) the map emulation problem. Rows and columns in (a) correspond to different tasks and QRs, respectively. The first column in (b) corresponds to the multiplexed QR.

Figure 5

The spatial multiplexing schematic. The same input sequence is injected into two distinct 5-qubit QRs. The internal states $\mathrm{Tr}({\rho }_l{Z}^{(i)})$ of the two QRs are linearly combined to form a single output.

Figure 6

Quantum circuit implementing the QND measurements by coupling ancilla qubits $|0{⟩}^{\otimes n}$ with the QR system qubits $|\psi {⟩}_{l-1}$.

Figure 7

Quantum circuits for the (a) 10-qubit QR and (b) 4-qubit QR on the Boeblingen device.

Figure 8

Quantum circuits for the (a) 5-qubit Ourense QR and (b) 5-qubit Vigo QR.

Figure 9

Full washout, train, and test input-output sequences for (a) the multistep ahead prediction problem and (b) the map emulation problem. The first rows in (a) and (b) show the input sequences.

Figure 10

The full target output sequences, the train and test output sequences of the four QRs for each task on the multistep ahead prediction problem. Each column corresponds to each $n$-qubit QR output and each row corresponds to each task.

Figure 11

The full target output sequences, the train and test output sequences of the QRs for each task on the map emulation problem. (a) The two train output sequences and (b) the test output sequence. The columns (from left to right) correspond to the multiplexed 5-qubit QRs, 5-qubit Ourense QR, and the 5-qubit Vigo QR. Each row corresponds to each task.

Figure 12

Input sequence, and experimental and simulation results for each qubit of the four QRs at each time step $l=1,\dots ,30$ for the multistep ahead prediction problem.

Figure 13

Experimental and simulation results for each qubit $i=0,\dots ,4$ and each time step $l=1,\dots ,24$ for the map emulation problem. Three input sequences are used in this problem, labeled as inputs I, II, and III. Row $i$ in a subfigure corresponds to the experimental data for the $i\mathrm{th}$ input sequence. Column $j$ corresponds to the experimental data for the $j\mathrm{th}$ qubit. (a) The experimental data for the 5-qubit Ourense QR. (b) The experimental data for the 5-qubit Vigo QR.