Analytical formulas for the performance scaling of quantum processors with a large number of defective gates

Removing a single logical gate from a classical information processor renders this processor useless. This is not so for a quantum information processor. A large number of quantum gates may be removed without significantly affecting the processor's performance. In this paper, focusing on the quantum Fourier transform (QFT) and quantum adder, we show even more: Even if most of its gates are eliminated and the remaining gates are selected from a randomly generated set, the QFT, one of the most useful quantum processors, and the quantum adder, one of the most basic building blocks of a universal quantum computer, still operate with satisfactory success probability, comparable to that of a quantum computer constructed with perfect gates. We support these conclusions by first laying out a general analytical framework and then deriving analytical scaling relations, which are in excellent agreement with our numerical simulations. The demonstrated robustness of the QFT and quantum adder, to the point where randomly generated quantum gates take the place of the exact gates, is an important boon for the construction of quantum computers, since it shows that stringent gate error tolerances do not have to be met to obtain satisfactory performance of the corresponding quantum processors. Our analytical techniques are powerful enough to generate asymptotic scaling laws for any gate defect model of quantum information processors and we illustrate this point by explicitly computing asymptotic analytical scaling formulas for several other defect models as well.

Figure 1

Logic circuit of a five-qubit QFT. Hadamard gates are denoted by H and phase rotation gates by ${\theta }_j$, whose rotation angle is $\pi /2^j$ [26], where $j$ is the qubit distance between control and target qubits.

Figure 2

Exact rotation angles ${\theta }_j$ (red closed squares) compared with their corresponding approximations ${\tilde{\theta }}_j$ (green closed circles) as a function of qubit distance $j$ for $j=1,...,10$. The approximations ${\tilde{\theta }}_j$ are the best matches for their corresponding ${\theta }_j$, drawn from a set of $N=20$ randomly generated rotation angles. The plateau in $\tilde{\theta }$ is a consequence of having small $N$, i.e., ${\theta }_j$ for $j\ge 7$ is approximated by the same approximate angle ${\tilde{\theta }}_j$. Although ${\theta }_j$ and ${\tilde{\theta }}_j$ appear to be close on a logarithmic scale, their percentage differences are large, exceeding $30\%$ from $j=7$ on. Nevertheless, a QFT equipped with these gates exhibits acceptable performance.

Figure 3

Fidelity $F$ of an eight-qubit QFT as a function of integer input state $|\alpha \rangle$. Each plot symbol is a result of averaging over 1000 realizations of the random set $S_N$, where $N=30$ was used in this figure. As the bit spectra of $\alpha$ become filled with 1's, the fidelity $F$ decreases. As expected, and proved in Appendix pp1, the case $\alpha =2^8-1$ (rightmost point in the figure), where all bits of $\alpha$ are 1, returns the worst $F$.

Figure 4

Fidelity $F$ of the QFT equipped with the randomly hierarchical rotation gates ${\tilde{\theta }}_j$ as a function of the number of qubits $n$. Shown are the cases of $N=20$ (red pluses), $N=40$ (green crosses), and $N=60$ (blue asterisks), each averaged over 100 ensembles of the respective $S_N$. (a) Full-bandwidth QFT and (b) banded QFT with bandwidth $b=8$.

Figure 5

Complement $1-F$ of the fidelity $F$ of the QFT equipped with the randomly hierarchical rotation gates ${\tilde{\theta }}_j$ as a function of the number of random gates $N$. In order to match the random bit spectrum assumption used in the analytical calculations, all numerical simulation data are averaged over all possible integer input states $|\alpha \rangle$, where $\alpha =0,...,2^n-1$, for an $n$-qubit QFT, in addition to the averaging over 100 realizations of $N$ random gates. (a) Full-bandwidth QFT and (b) banded QFT with bandwidth $b=4$. Shown are the cases with $n=5$, 6, 7, and 8, corresponding to pluses (red), crosses (green), asterisks (blue), and squares (purple), respectively.

Figure 6

Logic circuit of a five-qubit quantum Fourier adder in the Fourier space (blue box). A complete quantum adder circuit requires both a QFT and an inverse QFT operation, performed before and after application of the quantum Fourier adder (see the top of the figure). Phase rotation gates are represented by the ${\theta }_j$ boxes; their respective rotation angles are $\pi /2^j$ [26], where $j$ is the qubit distance between control and target qubits. The dashed lines represent the qubit lines of the addend $\nu$ of the adder. The solid lines represent the qubit lines of the computational qubits.

Figure 7

Fidelity $F$ of the quantum adder equipped with the randomly hierarchical rotation gates ${\tilde{\theta }}_j$ as a function of the number of qubits $n$. Shown are the cases of $N=20$ (red pluses), $N=40$ (green crosses), and $N=60$ (blue asterisks). Each plot symbol is a result of averaging over 10 000 ensembles of the random set $S_N$. (a) Full-bandwidth quantum adder and (b) banded quantum adder with bandwidth $b=8$.

Figure 8

Complement $1-F$ of the fidelity $F$ of the quantum adder equipped with the randomly hierarchical rotation gates ${\tilde{\theta }}_j$ as a function of the number of random gates $N$. In order to match the random bit spectrum assumption used in the analytical calculations, all numerical simulation data are averaged over all possible integer input states $|\alpha \rangle$, where $\alpha =0,...,2^n-1$ and all possible integer addends $\nu =0,...,2^n-1$. Shown are the results of averaging over 10 000 realizations of the random set $S_N$. (a) Full-bandwidth quantum adder and (b) banded quantum adder with bandwidth $b=4$. The cases with $n=5$, 6, 7, and 8 correspond to pluses (red), crosses (green), asterisks (blue), and squares (purple), respectively.

Figure 9

Complement $1-F$ of the fidelity $F$ of the full-bandwidth quantum adder equipped with Gaussian-defective rotation gates ${\tilde{\theta }}_j$ as a function of the standard deviation $\sigma$ of the gate errors. In order to match the random bit spectrum assumption used in the analytical calculations, all numerical simulation data are averaged over all possible integer input states $|\alpha \rangle$, where $\alpha =0,...,2^n-1$ and all possible integer addends $\nu =0,...,2^n-1$. Shown are the results of averaging over 10 000 realizations of the defects: (a) RC, (b) RU, (c) AC, and (d) AU. The cases with $n=5$, 6, 7, and 8 correspond to pluses (red), crosses (green), asterisks (blue), and squares (purple), respectively.

Figure 10

Complement $1-F$ of the fidelity $F$ of the banded quantum adder (bandwidth $b=4$) equipped with the Gaussian-defective rotation gates ${\tilde{\theta }}_j$ as a function of the standard deviation $\sigma$ of the gate errors. In order to match the random bit spectrum assumption used in the analytical calculations, all numerical simulation data are averaged over all possible integer input states $|\alpha \rangle$, where $\alpha =0,...,2^n-1$ and all possible integer addends $\nu =0,...,2^n-1$. Shown are the results of averaging over 10 000 realizations of the defects: (a) RC, (b) RU, (c) AC, and (d) AU. The cases with $n=5$, 6, 7, and 8 correspond to pluses (red), crosses (green), asterisks (blue), and squares (purple), respectively.

Figure 11

Complement $1-F$ of the fidelity $F$ of the full-bandwidth quantum adder equipped with uniformly defective rotation gates ${\tilde{\theta }}_j$ as a function of the defect strength parameter $\epsilon$. In order to match the random bit spectrum assumption used in the analytical calculations, all numerical simulation data are averaged over all possible integer input states $|\alpha \rangle$, where $\alpha =0,...,2^n-1$ and all possible integer addends $\nu =0,...,2^n-1$. Shown are the results of averaging over 10 000 realizations of the defects: (a) RC, (b) RU, (c) AC, and (d) AU. The cases with $n=5$, 6, 7, and 8 correspond to pluses (red), crosses (green), asterisks (blue), and squares (purple), respectively.

Figure 12

Complement $1-F$ of the fidelity $F$ of the banded quantum adder (bandwidth $b=4$) equipped with the uniform-defective rotation gates ${\tilde{\theta }}_j$ as a function of the defect strength parameter $\epsilon$. In order to match the random bit spectrum assumption used in the analytical calculations, all numerical simulation data are averaged over all possible integer input states $|\alpha \rangle$, where $\alpha =0,...,2^n-1$ and all possible integer addends $\nu =0,...,2^n-1$. Shown are the results of averaging over 10 000 realizations of the defects: (a) RC, (b) RU, (c) AC, and (d) AU. The cases with $n=5$, 6, 7, and 8 correspond to pluses (red), crosses (green), asterisks (blue), and squares (purple), respectively.

Figure 13

Exact rotation angles ${\theta }_j$ (red closed squares) compared with their corresponding approximations ${\tilde{\theta }}_j$ (green closed circles) as a function of qubit distance $j$ for $j=1,...,10$. The approximations ${\tilde{\theta }}_j$ are the best matches for their corresponding ${\theta }_j$, drawn from a set of $N=20$ random numbers whose statistical distribution is uniform on an exponential scale ranging from 0 to $n$. Shown in the figure is the case of $n=17$.

Figure 14

Exact rotation angles ${\theta }_j$ (red closed squares) compared with their approximations ${\tilde{\theta }}_j$ for the uniform (green closed circles) and exponential (blue closed triangles) hierarchies, as a function of qubit distance $j$ for $j=1,...,10$. Data are imported from Figs. 2 and 13 and plotted together on a linear scale for the convenience of the readers. The inset shows the averaged difference $\langle {\epsilon }_j\rangle =\langle |{\tilde{\theta }}_j-{\theta }_j|\rangle$ (green closed circles and blue closed triangles) for the uniform and exponential hierarchies, respectively, where averages were performed over 10 000 realizations of the random set $S_N$ with $N=20$ and $n=17$ for each hierarchy.