Realizing quantum linear regression with auxiliary qumodes

In order to exploit quantum advantages, quantum algorithms are indispensable for operating machine learning with quantum computers. We here propose an intriguing hybrid approach of quantum information processing for quantum linear regression, which utilizes both discrete and continuous quantum variables, in contrast to existing wisdoms based solely upon discrete qubits. In our framework, data information is encoded in a qubit system, while information processing is tackled using auxiliary continuous qumodes via qubit-qumode interactions. Moreover, it is also elaborated that finite squeezing is quite helpful for efficiently running the quantum algorithms in a realistic setup. Comparing with an all-qubit approach, the present hybrid approach is more efficient and feasible for implementing quantum algorithms, still retaining exponential quantum speed-up.

Figure 1

Illustration of the hybrid approach that exploits the best of both qubits and qumodes. In our implementation of quantum linear regression, qubits and qumodes are minimally coupled by connecting only to an ancillary qubit.

Figure 2

Quantum circuit for implementing a modified exponential swap gate. An ancillary qubit is prepared at $|+\rangle$. Qumodes and qubits are not directly coupled.

Figure 3

Ions (in red color) involved at the training and prediction steps.

Figure 4

Quantum circuit for preparing $|{\psi }_A\rangle$. Here $D_m=e^{i{\theta }_m{\sigma }_y}$ ($m=0,1,2,3$) with $\theta =arctan(a_1^{\left(m\right)}/a_0^{\left(m\right)})$ are single-qubit gates, which realize $D_m|0\rangle =a_0^{\left(m\right)}|0\rangle +a_1^{\left(m\right)}|1\rangle$. $H$ is the Hadamard gate. Two controls are added to $D_m$, such that they generate $|m\rangle \left(a_1^{\left(m\right)}\right|0\rangle +a^{\left(m\right)}|1\rangle )$, respectively. We have defined $|0\rangle \equiv |00\rangle ,|1\rangle \equiv |10\rangle ,|2\rangle \equiv |01\rangle ,|3\rangle \equiv |11\rangle$. The quantum circuit outputs the required state $|{\psi }_A\rangle ={\sum }_{m=0}^3{\sum }_{n=0}^1{a}_n^{\left(m\right)}|m\rangle |n\rangle$.

Figure 5

The relation of fidelity with squeezing $\frac{1}{s}$ (in brown) and with the regularization $\chi$ (in blue).

Figure 6

Fidelity with squeezing under different regularization $\chi =0.01,0.1$.