High-fidelity qubit measurement with a microwave-photon counter

High-fidelity, efficient quantum nondemolition readout of quantum bits is integral to the goal of quantum computation. As superconducting circuits approach the requirements of scalable, universal fault tolerance, qubit readout must also meet the demand of simplicity to scale with growing system size. Here we propose a fast, high-fidelity, scalable measurement scheme based on the state-selective ring-up of a cavity followed by photodetection with the recently introduced Josephson photomultiplier (JPM), a current-biased Josephson junction. This scheme maps qubit state information to the binary digital output of the JPM, circumventing the need for room-temperature heterodyne detection and offering the possibility of a cryogenic interface to superconducting digital control circuitry. Numerics show that measurement contrast in excess of 95% is achievable in a measurement time of 140 ns. We discuss perspectives to scale this scheme to enable readout of multiple qubit channels with a single JPM.

Figure 1

(a) Schematic diagram of the Josephson photomultiplier (JPM) circuit. The junction is biased with a dc current, and microwaves are coupled to the junction via an on-chip capacitor. In the simplest implementation, switching of the junction creates a voltage pulse that is read out by a room-temperature comparator circuit. (b) Junction potential energy landscape. The junction is initialized in the ground state $|g\rangle$. An incident photon induces a transition to the first excited state $|e\rangle$, which rapidly tunnels to the continuum with rate ${\gamma }_{\mathrm{J}}$. (c) Counter-based measurement in cQED. “Bright” and “dark” cavity pointer states result in binary digital output from the JPM: “click” or “no click.” (d) In-phase (I) and quadrature (Q) phase space portrait of the cavity state after the ring-up, highlighting pointers to the $|0\rangle$ state in red and to the $|1\rangle$ state in blue.

Figure 2

Schematic of the three-stage measurement protocol: the upper panel describes the relevant control, while the lower panel represents the corresponding cavity state. (a) In the drive stage, the cavity is driven strongly and coherently at the cavity frequency dressed by the qubit $|1\rangle$ state, ${\omega }_{\mathrm{d}}={\omega }_1={\omega }_{\mathrm{C}}-{\chi }_{\mathrm{Q}}+{\chi }_{\mathrm{J}}$, for a duration $t_{\mathrm{d}}=\pi /{\chi }_{\mathrm{Q}}$ (assuming a square pulse). This projects the qubit onto either $|0\rangle$ or $|1\rangle$ and conditionally populates the cavity with a large number of photons $n\sim |{\alpha }_1{|}^2$ when the qubit is projected onto the $|1\rangle$ state. (b) During the measurement phase, the JPM is tuned into resonance with the cavity and allowed to interact; a bright cavity switches the JPM to its voltage state while a dark cavity leaves the JPM in the supercurrent state. This conditionally squeezes the cavity state by a small amount (not shown here). (c) In the reset stage, the cavity is again driven coherently at ${\omega }_{\mathrm{d}}$, conditionally displacing the cavity to a near-vacuum state.

Figure 3

Cavity occupation as a function of the duration of the applied drive, $t_{\mathrm{d}}$, for the qubit in states $|0\rangle$ and $|1\rangle$. Here ${\chi }_{\mathrm{Q}}/\pi =10$ MHz, such that the optimal drive time is $t_{\mathrm{d}}=100$ ns.

Figure 4

(a) Detection probability $P\left(\right|\alpha {|}^2,t_{\mathrm{m}})$ as a function of measurement time $t_{\mathrm{m}}$ (horizontal axis) and cavity occupation ${\left|\alpha \right|}^2$ (color) for system parameters $g_{\mathrm{J}}/2\pi =50$ MHz, ${\gamma }_{\mathrm{J}}=200$ MHz, ${\gamma }_{\mathrm{D}}=1$ MHz, and ${\gamma }_{\mathrm{R}}=200$ MHz. (b) Detection probability $P\left(\right|\alpha {|}^2,t_{\mathrm{m}})$ as a function of cavity occupation ${\left|\alpha \right|}^2$, both numerical simulations and analytic fit, for $t_{\mathrm{m}}=25$ and 50 ns.

Figure 5

(a) Excited qubit detection probability $P\left(\right|{\alpha }_1{|}^2,t_{\mathrm{m}})$, ground qubit detection probability $P\left(\right|{\alpha }_0{|}^2,t_{\mathrm{m}})$, and measurement contrast versus measurement time. Here $|{\alpha }_1{|}^2=10$ and $|{\alpha }_0{|}^2=0$, with system parameters as before. (b) Measurement contrast as a function of bright-count rate ${\gamma }_{\mathrm{J}}$, for various $|{\alpha }_1{|}^2$ and $t_{\mathrm{m}}=50$ ns. The relaxation rate remains fixed at ${\gamma }_{\mathrm{R}}=200$ MHz, while the dark-count rate changes such that the ratio ${\gamma }_{\mathrm{J}}/{\gamma }_{\mathrm{D}}=200$ is unchanged.

Figure 6

Numerically evaluated overlap error for the reset pulse described in the text. The number of photons $N$ removed from the cavity is shown in the legend.

Figure 7

(a) Qubit ${\sigma }_z$ expectation for the qubit initially in the excited state versus measurement time (Jaynes-Cummings Hamiltonian) and (b) process fidelity of qubit readout for both the dispersive Hamiltonian and the Jaynes-Cummings Hamiltonian. Coupling strength and tunneling rates are as used throughout. Here $|{\alpha }_0{|}^2=0$ and $|{\alpha }_1{|}^2=9$.

Figure 8

(a) Cavity occupation during the drive stage for the full Jaynes-Cummings Hamiltonian. (b) Measurement contrast for the dispersive Hamiltonian and the Jaynes-Cummings Hamiltonian. As elsewhere, ${\chi }_{\mathrm{Q}}/\pi =10$ MHz, $g_{\mathrm{J}}/2\pi =50$ MHz, ${\gamma }_{\mathrm{J}}=200$ MHz, ${\gamma }_{\mathrm{D}}=1$ MHz, and ${\gamma }_{\mathrm{R}}=200$ MHz. In both plots, the drive strength is chosen such that $|{\alpha }_1{|}^2=9$ for $t_{\mathrm{d}}=100$ ns for the dispersive Hamiltonian.