Quantum Period-Finding using One-Qubit Reduced Density Matrices

Abstract

The quantum period-finding (QPF) algorithm can compute the period of a function exponentially faster than the best-known classical algorithm. In standard QPF, the output state has a primary contribution from $r$ high-probability bit strings, where $r$ is the period. Measurement of this state, combined with continued fraction analysis, reveals the unknown period. Here, we explore a different approach to QPF, where the period is obtained from single-qubit quantities $-$ specifically, the set of one-qubit reduced density matrices (1-RDMs) $-$ rather than the output bit strings of the entire quantum circuit. Using state-vector simulations, we compute the 1-RDMs of the QPF circuit for a generic periodic function. Analysis of these 1-RDMs as a function of period reveals distinctive patterns, which allows us to obtain the unknown period from the 1-RDMs using a numerical root-finding approach. Our results show that the 1-RDMs $-$ a set of $O(n)$ one-qubit marginals $-$ contain enough information to reconstruct the period, which is typically obtained by sampling the space of $O(2^n)$ bit strings. Conceptually, this can be viewed as a "compression" of the information in the QPF algorithm, which enables period-finding from $n$ one-qubit marginals. Our results motivate the development of approximate simulations of reduced density matrices to design novel period-finding algorithms.

Figures (5)

• Figure 1: Quantum circuit for QPF, shown for $n=3$ qubits in both registers. The oracle prepares the periodic function $f$ in the second register, shown at the bottom in red. Measuring the second register puts the first register in state $|\varphi(r)\rangle$, which becomes $|\Psi(r)\rangle$ after the quantum Fourier transform (QFT) (see text for definitions). The final measurement, combined with continued fraction analysis, provides the period $r$.
• Figure 2: Computed diagonal elements of the 1-RDMs for the QPF circuit, expressed as $a_z^{(q)}(r) = \rho^{(q)}_{00}(r) - 0.5$, shown as a function of period $r$ for each qubit $q$. We show results for a) $n=3$ and b) $n=5$ qubits. The periods with $a^{(q)}_z(r)\!>\!0$ fit the general rule in Eq. \ref{['eq:peaks']}. For qubit $q=0$, a nonzero value of $a_z^{(q)}(r)$ occurs only for periods $r\!=\!2^k$. For each subsequent qubit ($q=1,2,...,n-1$), there are new periods with $a^{(q)}_z(r)\! >\! 0$, shown with colors not used in previous qubits, and removes some periods that had $a_z\!>\!0$ in preceding qubits. The last qubit, $q\!=\!n-1$, has $a_z(r) > 0$ only for odd periods.
• Figure 3: Comparison of approximate versus exact $a^{(q)}_z(r)$ for $n\!=\!6$ qubits, shown for a) the last qubit, $q=n-1$, and b) the penultimate qubit, $q=n-2$. These cases correspond to $q'\!=\!0$ and $q'\!=\!1$, respectively. The first quarter of the domain (periods smaller than $N/4$), where the approximation is more accurate, is shown with a shaded area in both plots.
• Figure 4: Accuracy for quantum period-finding from 1-RDMs using our numerical approach. Results are shown for $N\!=\!64,~128,~256$, which corresponds to $n\!=\!6,~7,~8$ qubits respectively, by plotting the accuracy as a function of the number of extra qubits added to improve the root-finding procedure.
• Figure 5: Relation between the range $\mathcal{R}(\beta_q)$ and the values $jN/r$ for a) the last qubit $q\!=\!n-1$, for $r\!=\!7$, and b) the next to last qubit, $q\!=\!n-2$, for $r\!=\!2r'$, with $r'\!=\!7$. The range of real values spanned by the strings in $\beta_q$, $\mathcal{R}(\beta_q)$, is a set of continuous intervals, shown using blue color. The vertical dashed lines represent $jN/r$, for $j=0,1,...,r-1$, with the values overlapping with $\mathcal{R}(\beta_q)$ highlighted in red.