Effects of dynamical phases in Shor’s factoring algorithm with operational delays

Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist idle finite time intervals between consecutive operations in a realistic quantum computing process. During these delays, coherent errors will accumulate from the dynamical phases of the superposed wave functions. Here we explore the sensitivity of Shor’s quantum factoring algorithm to such errors. Our results clearly show a severe sensitivity of Shor’s factorization algorithm to the presence of delay times between successive unitary transformations. Specifically, in the presence of these coherent errors, the probability of obtaining the correct answer decreases exponentially with the number of qubits of the work register. A particularly simple phase-matching approach is proposed in this paper to avoid or suppress these coherent errors when using Shor’s algorithm to factorize integers. The robustness of this phase-matching condition is evaluated analytically and numerically for the factorization of several integers: 4, 15, 21, and 33.

Figure 1

Quantum circuit for implementing Shor’s algorithm with time delays ${\tau }_j^{+}(j=1,2,3)$ between the successive operations. Here $H$ refers to a Hadamard gate, while $F$ refers to a quantum Fourier transformation. Each block operation is assumed to be exactly performed in a very short time interval $\epsilon$ (so that phases accumulated during the operations are either accounted for by the operations themselves or simply neglected).

Figure 2

The probability $P\left(k\right)$ [see Eq. (6)] of observing values of $k$ for different values of $\tau \mathrm{\Delta }=({ϵ}_1-{ϵ}_0)\tau =0$, $0.4\pi$, $\pi$, $1.6\pi$, and $2\pi$, given $N=21$, $q=512$, $a=5$, and the expected order $r=6$. Here, $\tau$ is the total effective delay time between unitary operations. The correct outputs are obtained when the phase-matching condition $\tau \mathrm{\Delta }=2\pi$ (or the ideal case $\tau \mathrm{\Delta }=0$) is satisfied. The probabilities of obtaining the correct outputs far from the phase-matching conditions are very low. (See the second, third, and fourth panels. Note the different scales for the vertical axes.) Indeed, as shown in the bottom three panels, many incorrect results are produced when the phase-matching condition given by Eq. (7) is not enforced.

Figure 3

The probability $P_e$ of obtaining the correct results versus $\mathrm{\Delta }\tau =({ϵ}_1-{ϵ}_0)\tau$ for running Shor’s factoring algorithm in the presence of delays. The lines for $r=4$, 6, 10 correspond to the cases where 4, 9, 11 work qubits, given $q=16$, 512, 2048, are used to factorize $N=15$, 21, 33 with $a=13$, 5, 5, respectively. Note that the expected outputs can be obtained at the phase-matching points: $\mathrm{\Delta }\tau =2\pi ,4\pi$.

Figure 4

The probability $p_e^{\left(L\right)}\left(k_e\right)$ of obtaining one of the correct results versus the number $L$ of work qubits used to run the quantum algorithm factorizing $N=15$ in the presence of a delay $\mathrm{\Delta }\tau =({ϵ}_1-{ϵ}_0)\tau =5\pi ∕3$. The straight line shows that this probability $p_e^{\left(L\right)}\left(k_e\right)$ decreases exponentially with the number $L$ of qubits used. The points on the line show the probability of obtaining one of the correct outputs $k_e=(0,4,8,12)$ for four-qubit, (0, 8, 16, 24) for five-qubit, (0, 16, 32, 48) for six-qubit, (0, 32, 64, 96) for seven-qubit, and (0, 64, 128, 192) for eight-qubit cases, respectively.

Figure 5

The probabilities $P_e$ (for factorizing $N=15$ using eight work qubits) of obtaining the correct results for different phase-matching cases: $⟨\mathrm{\Delta }⟩\tau =2\pi$, $4\pi$, $6\pi$, $8\pi$, with a common Gaussian energy splitting fluctuation with $\sigma ∕⟨\mathrm{\Delta }⟩=0.5\mathrm{\%}$. Note that this probability $P_e$ is higher at the phase-matching points with shorter total delay time $\tau$.

Figure 6

The probabilities $P_e$ (for factorizing $N=15$ using eight work qubits) of obtaining the correct results for different fluctuations of energy splittings: $\sigma ∕⟨\mathrm{\Delta }⟩=0.01\mathrm{\%}$, 0.3%, 0.7%, 1.1%, with a common phase-matching point: $⟨\mathrm{\Delta }⟩\tau =2\pi$. Note that the probability at the phase-matching point is still sufficiently high, even if the energy splittings of the qubits exist with certain fluctuations around the average value $⟨\mathrm{\Delta }⟩$.

Figure 7

Quantum circuits formed by the elementary single- and two-qubit logic gates for performing (a) Hadamard gate for one qubit and (b) a quantum Fourier transformation for three-qubit. Here, ${\delta }_l(l=1,2,\mathrm{\dots })$ and ${\rho }_{kl}(k=0,1,2,\mathrm{\dots })$ refer to the operational delays inside them, respectively. In the logical basis, the single-qubit gate ${\widehat{R}}_z=\exp (i\pi {\sigma }_z∕4)$ and the two-qubit controlled-phase gate $R_k=\mid 00⟩⟨00\mid +\mid 01⟩⟨01\mid +\mid 10⟩⟨10\mid +\exp (2i\pi ∕2^k)\mid 11⟩⟨11\mid$ are diagonal, while the single-qubit ${\widehat{R}}_x=\exp (i\pi {\sigma }_x∕4)$ is not.