Heisenberg-limited Bayesian phase estimation with low-depth digital quantum circuits

TL;DR

This work develops a scheme that achieves near-optimal precision up to a constant overhead for Bayesian phase estimation, using simple digital quantum circuits with depths scaling logarithmically with the number of qubits.

Abstract

Optimal phase estimation protocols require complex state preparation and readout schemes, generally unavailable or unscalable in many quantum platforms. We develop and analyze a scheme that achieves near-optimal precision up to a constant overhead for Bayesian phase estimation, using simple digital quantum circuits with depths scaling logarithmically with the number of qubits. We find that for Gaussian prior phase distributions with arbitrary widths, the optimal initial state can be approximated with products of Greenberger-Horne-Zeilinger states with varying number of qubits. Using local, adaptive measurements optimized for the prior distribution and the initial state, we show that Heisenberg scaling is achievable and that the proposed scheme outperforms known schemes in the literature that utilize a similar set of initial states. For an example prior width, we present a detailed comparison and find that is also possible to achieve Heisenberg scaling with a scheme that employs non-adaptive measurements, with the right allocation of copies per GHZ state and single-qubit rotations. We also propose an efficient phase unwinding protocol to extend the dynamic range of the proposed scheme, and show that it outperforms existing protocols by achieving an enhanced precision with a smaller number of additional atoms. Lastly, we discuss the impact of noise and imperfect gates.

Figures (15)

• Figure 1: Summary of contributions. We propose a scheme that, given the prior width $\delta\phi$ and the total number of qubits $N$, computes the optimal partition over blocks of GHZ states, as well as optimal local, adaptive measurements. Partitions of blocks of GHZ states can be used to approximate the optimal initial states for phase estimation. If optimal measurements are available, blocks of GHZ states approximately reach the benchmark sensitivity for all prior widths. If local, adaptive measurements are used instead, we obtain Heisenberg scaling up to a constant overhead. Furthermore, optimal partitions can be rescaled to include slow atoms (atoms that accumulate phases of $\phi/2, \phi/4, \dots$), which enable an extended dynamic range beyond $\delta\phi \gg \pi$. We show that the proposed scheme surpasses other phase unwinding protocols in the literature that extend the dynamic range.
• Figure 2: Illustration of the proposed scheme. Summary of the introduced protocol for $N=4$ qubits, and a prior width of $\delta\phi = \pi/3 \approx 1.05$ rad. (a) The possible partitions of the blocks of GHZ states for $N=4$ qubits are given in the Table, and the MSE as a function of $\phi$ is plotted. For this prior width, the best-performing partition is plotted in yellow, highlighted in the Table. (b) Optimization over local, adaptive measurements. The measurement consists of successive $\hat{Z}(\Phi_i)$ operations and projective measurements. The $\hat{Z}(\Phi_i)$ operation corresponds to a rotation around the $z$-axis by $\Phi_i$ in the Bloch sphere picture, such that $\phi-\Phi_i$ is measured. Based on the measurement outcome, denoted with $\ket{+}$ and $\ket{-}$, the next rotation angle is determined. During the optimization, $m$ phases are sampled uniformly from the interval $[-6 \delta\phi,\, 6\delta\phi]$, which are then used to compute the Bayesian mean squared error (BMSE) analytically. Gradient descent is used to tune the single-qubit rotations $\Phi_i$ in the adaptive measurement. This process is iterated until the BMSE converges to a minimum. The path highlighted with orange corresponds to the measurement outcome $\{+,-,+\}$, for example. (c) Numerical values (in units of radians) of the single-qubit rotations $\Phi_i$ at the end of the optimization.
• Figure 3: Performance of the proposed scheme for $N = 21$ qubits. (a) Root Bayesian mean squared error (RBMSE) $\Delta\tilde{\phi}$ as a function of the prior width, $\delta\phi$, for $N = 21$. The performance of the phase estimation with a 21-qubit GHZ state, 21-qubit CSS, the OQI, and the proposed scheme is plotted in blue, yellow, gray, and red, respectively. The different shapes of the red markers correspond to different ways of partitioning the 21 qubits into blocks of GHZ states, listed in the table in b. (b) Optimal partitions of 21 qubits into blocks of GHZ states for different prior widths. For a small prior width (first row), we see that the optimal strategy is to use the GHZ state with the maximum number of atoms possible, which is the 16-atom GHZ state for $N = 21$. The remaining atoms are distributed to smaller GHZ states. In contrast, for a large prior width within the dynamic range (last row), it is more advantageous to use GHZ states with smaller numbers of atoms and distribute them equally, i.e. use 3 blocks of each. (c) The amplitudes of states with $n_{\text{up}}$ excitations, $0 \leq n_{\text{up}} \leq 21$, for the optimal initial states of the OQI and the initial states obtained with blocks of GHZ states, for different prior widths. $n_{\text{up}} = 0$ ($n_{\text{up}} = 21$) corresponds to the state $\ket{0}^{\otimes N}$($\ket{1}^{\otimes N}$). We see that for $\delta\phi \approx 1$ rad, the optimal initial state of the OQI resembles a sine state. (d) MSE of the OQI, CSS, and the proposed scheme for $N = 21$ and $\delta \phi = 0.7$ rad, plotted in purple, yellow, and red, respectively. The proposed scheme achieves a metrological gain of $g=2.29$ (3.59 dB).
• Figure 4: Performance as a function of the number of qubits $N$. The root Bayesian mean squared error (RBMSE) as a function of number of qubits $N$, normalized by the prior width, for $\delta\phi = 0.7$ rad. We see that the CSS and the OQI constitute the SQL and HL, respectively. For larger qubit numbers, the BMSE of all of the schemes converge to a constant independent of $N$. We also plot the $\pi$-corrected HL in absence of phase slip errors, which touches the OQI curve at $N \approx 70$. We observe that all of the schemes that use blocks of GHZ states show sub-SQL sensitivity. Furthermore, the scheme with a varying block size higgins_demonstrating_2009 and the proposed scheme obtain HS. For all $N$, the proposed scheme obtains the maximal sensitivity, among existing schemes. Inset. We plot the RBMSE of the OQI, CSS, and the proposed scheme in the range $40 \leq N \leq 100$, with gray, yellow, and red, respectively. We see that the proposed scheme shows HS with an overhead of $\approx 1.56$, shown with the light green line.
• Figure 5: Extending the dynamic range with slow atoms. Root Bayesian mean squared error (RBMSE) $\Delta\tilde{\phi}$, normalized by $2\pi$, as a function of the total qubit number $N$, for different prior widths. For small enough prior widths ($\delta\phi = 0.7$ rad), no slow atoms are needed. The RBMSE of the proposed scheme in this regime is plotted in gray. For larger prior widths, the number of atoms needed to extend the dynamic range depends on the phase unwinding protocol. These protocols estimate $\phi = 2\pi P + \theta$, $P \in \mathbb{Z}, \, \theta = \phi \,\text{mod}\,2\pi$. We define non-adaptive (estimates $P$ with slow atoms and $\theta$ with the scheme with a varying block size higgins_demonstrating_2009) and adaptive (estimates both $P$ and $\theta$ with slow atoms with, then estimates $\phi-\phi_\text{est}$ with the scheme with a varying block size) as phase unwinding. We also propose a more efficient phase unwinding protocol that extends the proposed scheme to large prior widths. Furthermore for the proposed scheme, in the limit of $N \gg 1$, the RBMSE for different prior widths converge to the RBMSE for $\delta\phi = 0.7$ rad. Therefore, in this limit, we can obtain a clock stability that scales as the total interrogation time $\tau$ even in the regime where $\tau \gamma_{\text{LO}} > 1$, where $\gamma_{\text{LO}}$ is the linewidth of the free running laser. Inset. MSE for the proposed phase unwinding scheme for $N = 16$ atoms and a prior width of $\delta\phi = 2.8$ rad. We see that the MSE is close to that of the OQI, and that the proposed scheme has a dynamic range of $\approx 8\pi$.