Hybrid quantum linear equation algorithm and its experimental test on IBM Quantum Experience

Abstract

We propose a hybrid quantum algorithm based on the Harrow-Hassidim-Lloyd (HHL) algorithm for solving a system of linear equations. In this paper, we show that our hybrid algorithm can reduce a circuit depth from the original HHL algorithm by means of a classical information feed-forward after the quantum phase estimation algorithm, and the results of the hybrid algorithm are identical to those of the HHL algorithm. In addition, it is experimentally examined with four qubits in the IBM Quantum Experience setups, and the experimental results of our algorithm show higher accurate performance on specific systems of linear equations than that of the HHL algorithm.

Figure 1

The circuit diagram for the HHL algorithm: the circuit consists of the QPE part, the AQE part and the inverse QPE part. The unitary gate $U_{\hat{A}}=e^{2\pi i\hat{A}}$ is used for controlled-unitary gates between the register R and the input qubit V while the controlled AQE indicates a set of controlled gates between the register R and the ancillary qubit A.

Figure 2

(a) Fidelities between solutions of the linear systems of equations in Eq. (10) and output states obtained from the HHL algorithm with k-qubit register for k = 1, 2, 3. (b) The probability distribution for measurement outcomes: The QPEA with 2-qubit register is performed on ${\hat{A}}_{\lambda }$ and $\overrightarrow{b}$ in Eq. (10). Since the QPEA makes use of 2 qubits as register, its measurement outcomes are two-bit strings b₁b₂ with b₁, b₂ ∈ {0, 1}, and Pr(b₁b₂) denotes the probability that the outcome is b₁b₂. Details of the probabilities are presented in Eqs (12) and (13).

Figure 3

The circuit diagrams of the original and hybrid HHL algorithms for ${\hat{A}}_{\lambda }$ and $\overrightarrow{b}$ in Eq. (10). (a) The controlled $U_{\lambda }^m$ gate, where $U_{{\hat{A}}_{\lambda }}^m={(e^{2\pi i{\hat{A}}_{\lambda }})}^m$ for $m\in \mathbb{Z}$. (b) The inverse quantum Fourier transform for two qubits. (c) Additional measurement devices to check outputs of the algorithms. In the hybrid HHL algorithm, AQE′ indicates a reduced AQE part (aqua color). The detailed circuit implementations of (a,b), and AQE′ are given in Fig. 4.

Figure 4

The circuit implementations on IBMQX setups: (a) the controlled $U_{\lambda }^m$ gate where $U_{{\hat{A}}_{\lambda }}^m={(e^{2\pi i{\hat{A}}_{\lambda }})}^m$ for $m\in \mathbb{Z}$, (b) the inverse quantum Fourier transform for two qubits, (c) the original AQE part, and (d) the reduced AQE part for the reduced HHL part when λ = 1/4, 2/4. Here, θ′₃: = θ₃−(θ₁ + θ₂).

Figure 5

Experimental results on IBMQX4: (a) the QPEA with 2-qubit register. (b) the original HHL algorithm and reduced HHL part.

Figure 6

The circuit diagram of the QPEA with n-qubit register: H and $F{T}^{†}$ are the Hadamard gate and the inverse quantum Fourier transform. At the end, every register qubit is measured in observable Z.