Resource Efficient Boolean Function Solver on Quantum Computer

TL;DR

A W-cycle circuit construction introduces a recursive idea to increase the solvable number of boolean equations given a fixed number of qubits, and a greedy compression technique is proposed to reduce the oracle circuit depth.

Abstract

Nonlinear boolean equation systems play an important role in a wide range of applications. Grover's algorithm is one of the best-known quantum search algorithms in solving the nonlinear boolean equation system on quantum computers. In this paper, we propose three novel techniques to improve the efficiency under Grover's algorithm framework. A W-cycle circuit construction introduces a recursive idea to increase the solvable number of boolean equations given a fixed number of qubits. Then, a greedy compression technique is proposed to reduce the oracle circuit depth. Finally, a randomized Grover's algorithm randomly chooses a subset of equations to form a random oracle every iteration, which further reduces the circuit depth and the number of ancilla qubits. Numerical results on boolean quadratic equations demonstrate the efficiency of the proposed techniques.

Figures (15)

• Figure 1: An $8$-qubit example as the illustration of the randomized Grover's algorithm. From left to right, top to bottom, the snapshots are the probability distribution among all states after $0$, $2$, $4$, and $6$ iterations. We observe that the probability distribution is concentrated more and more on the correct states during these $6$ iterations. The $x$-axis and $y$-axis represent the first and the last $4$ qubits respectively. The $z$-axis is the probability of the corresponding state. Note that for visual clarity, the $x$-axis labels and $y$-axis labels are not fully listed. The bottom bar chart shows the change in the sum of probability for correct states at each iteration. The sum reaches the maximum at $6$ iterations.
• Figure 2: Diagrams of basic quantum gates in quantum circuits. The X in the box represents a NOT gate. For CNOT and CCNOT, the solid dots are on the control qubits, and $\oplus$ is on the target qubit. The Z in the box represents a Z gate, whereas the MCZ gate is represented by solid dots on all the qubits it acts on. A toy example with an initial state $\ket{000}$ is given below the circuit.
• Figure 3: Grover's algorithm. In vanilla Grover's algorithm, all $G_i$s are $G = WO$ for $O$ and $W$ being defined in \ref{['eq:oracle-action']} and \ref{['eq:householder-action']} respectively.
• Figure 4: The effect of an iteration of Grover's algorithm. From the initial state $\ket{\psi}$, the oracle $O$ flips the sign of the solution to $g(x)=1$, as shown in \ref{['eq:oracle-action']}. Then, the diffusion operation $W$ makes a reflection of the state $O\ket{\psi}$ by $\ket{\psi}$. This single step of iteration rotates $\ket{\psi}$ to $G\ket{\psi}$ by an angle of $\theta$. This step reduces the probability of any state in $\ket{\alpha}$ (i.e., wrong solution) being the outcome of a measurement, which is $|\braket{\psi | \alpha}|^2$. Note that rotating over $\ket{\beta}$, i.e., repeating the iteration more than $K$ times computed in \ref{['eq:GroverIterNum']}, will contradictorily increase the probability of any state in $\alpha$ being the outcome of a measurement.
• Figure 5: Oracle circuit for boolean equation: $f(x) = x_1 \oplus x_1 x_2 =0$. The extra X gate ensures the ancilla in state $\ket{1}$ if $f(x)=0$.

Theorems & Definitions (5)

• Theorem 1
• proof
• Corollary 1
• Theorem 2
• Corollary 2