Numerical analysis of quantum circuits for state preparation and unitary operator synthesis

We perform optimal-control-theory calculations to determine the minimum number of two-qubit controlled-not (cnot) gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that all produce the desired result, even at the minimum number of gates. We also show that the number of entangling gates can be reduced if we use multiqubit entangling gates instead of two-qubit cnot gates, as one might expect based on parameter counting calculations. In addition to treating the general case of arbitrary target states or unitary operators, we apply the numerical approach to the special case of synthesizing the multiqubit Toffoli gate. This approach can be used to investigate any other specific few-qubit task and provides insight into the tightness of different bounds in the literature.

Figure 1

Number of independent parameters in a four-qubit quantum state (4QQS) (horizontal cyan line) and the number of independent parameters in a quantum circuit as a function of circuit size $N$. The red circles, green squares, and blue triangles correspond to using two-, three-, and four-qubit cz gates, respectively, along with single-qubit rotations in constructing the quantum circuit. The number of independent quantum circuit parameters obviously increases with circuit size. The lower bound for the circuit size needed to perfectly prepare an arbitrary four-qubit state is defined by the minimum value of $N$ for which the data point lies above the horizontal line.

Figure 2

Maximum achievable fidelity $F$ for $n$-qubit state preparation when using a quantum circuit that contains $N$ entangling gates. The green open circle corresponds to preparing a two-qubit state (i.e., $n=2$) using two-qubit cz gates (i.e., $m=2$). The magenta crosses and cyan diamonds are for the case $n=3$ and correspond to using two- and three-qubit cz gates, respectively. The red dots, green squares and blue pluses are for the case $n=4$ and correspond to using two-, three-, and four-qubit cz gates, respectively. We used ten different randomly generated target states for each setting. To show the statistical spread in the results, the red dots show the results for all ten instances of the target state for $n=4$ and $m=2$. For all other data points, we took the lowest value of maximum fidelity among the ten random instances. For a two-qubit system, a single cz gate is sufficient for perfect state preparation. For all other data sets, the last shown (i.e., largest-$N$) data point in each set has an infidelity $1-F\sim {10}^{-7}$ or less after ${10}^3$ optimization iterations and shrinks to below ${10}^{-12}$ if we continue the optimization to ${10}^6$ iterations, which implies that the small numbers that we obtain for $1-F$ are numerical errors, and the actual achievable fidelity in each one of these cases is $F=1$.

Figure 3

Histograms of fidelity $F$ values for all the possible cnot gate configurations for one instance of the state preparation problem with a four-qubit target state and using two-qubit cnot gates as the entangling gates. A logarithm function is used for the $x$ axis to magnify the region close to $F=1$ and make the features of the histogram in this region easier to discern. The blue dotted, green dashed, and red solid lines correspond to $N=4$, 5, and 6, respectively. The inset shows the high-fidelity tail of the $N=6$ histogram. The peak at $1-F={10}^{-12}$ includes all the configurations that gave $1-F<{10}^{-12}$, which turn out to be 20% of all possible configurations.

Figure 4

Correlations between fidelity values for quantum circuits of different sizes, specifically quantum circuits of size $N_1=2$ and those of size $N_2=5$. Each gate configuration (to which we refer as $Q_i$) in the set of all configurations with size $N_1$ produces one point in the plot. The $x$ axis shows the fidelity $F_1$ for the quantum circuit $Q_i$. The $y$ axis shows the maximum fidelity $F_2$ that can be achieved with a quantum circuit of size $N_2$ and containing $Q_i$ as its initial part. While there is some correlation between $F_1$ and $F_2$, the highest value of $F_2$ does not correspond to the highest value of $F_1$, indicating that optimizing the quantum circuit one layer at a time does not produce the global optimum among large quantum circuits.

Figure 5

Maximum achievable fidelity $F$ for three-qubit unitary operator synthesis as a function of quantum circuit size $N$. The red circles and green squares correspond to using two- and three-qubit cz gates, respectively. Randomly generated unitary operators were used as target operators. The inset shows the logarithm of the infidelity ${log}_{10}(1-F)$ for the high-fidelity points. The arrows indicate that $F$ can be made arbitrarily close to $F=1$ for the last point in each data set by increasing the number of optimization iterations, while other data points do not experience significant changes with an increased number of iterations.

Figure 6

Histograms of fidelity $F$ values for all the possible cnot gate configurations for the problem of synthesizing a three-qubit unitary operator using two-qubit cnot gates and single-qubit gates. The olive, green, orange, blue, magenta, cyan, and red lines correspond to $N=8$, 9, 10, 11, 12, 13, and 14, respectively. All the data in this figure were obtained using ${10}^3$ optimization iterations.

Figure 7

Maximum achievable fidelity $F$ as a function of quantum circuit size $N$ for synthesizing a multiqubit Toffoli gate. The red circles are for the well-known case of synthesizing a three-qubit Toffoli gate from two-qubit cnot gates. The green squares are for the case of synthesizing a four-qubit Toffoli gate from two-qubit cnot gates. The blue triangles are for the case of synthesizing a four-qubit Toffoli gate from three-qubit Toffoli gates.