Scalable two- and four-qubit parity measurement with a threshold photon counter

Parity measurement is a central tool for many quantum information processing tasks. In this work, we propose a method to directly measure two- and four-qubit parity with low overhead in hardware and software while remaining robust to experimental imperfections. Our scheme relies on dispersive qubit-cavity coupling and photon counting that is sensitive only to intensity; both ingredients are widely realized in many different quantum computing modalities. For a leading technology in quantum computing, superconducting integrated circuits, we analyze the measurement contrast and the back action of the scheme and show that this measurement comes close enough to an ideal parity measurement to be applicable to quantum error correction.

Figure 1

(a) Schematic of the experimental design showing the cavity, $N$ coupled qubits, and the photon counter (a JPM in this case). (b) and (c) Illustration of the dispersive shifts on the cavity for two- and four-qubit computational basis states.

Figure 2

Cavity photon number as a function of the length of the drive pulse $t_{\mathrm{D}}$ for the four-qubit parity bands. ${\chi }_{\mathrm{Q}}/\pi =10$ MHz, so the optimal drive time is $t_{\mathrm{D}}=100$ ns, which for $|{\alpha }_{\mathrm{O}}{|}^2=9$ sets $a_0=0.06$ MHz.

Figure 3

Bright-count probability for the four-qubit odd-parity bands, dark-count probability for even parity, and measurement contrast as functions of measurement time. The JPM bright count rate, inelastic relaxation rate, and dark-count rate are ${\gamma }_{\mathrm{J}}=200$ MHz, ${\gamma }_{\mathrm{R}}=200$ MHz, and ${\gamma }_{\mathrm{D}}=1$ MHz. The cavity-JPM coupling is $g_{\mathrm{J}}/2\pi =50$ MHz.

Figure 4

Even-parity detection probability, odd-parity coherence, and even-parity coherence as functions of the ${\chi }_{\mathrm{Q}}$ mismatch for two-qubit parity measurement. $\epsilon /{\chi }_{\mathrm{Q}}$ ranges from 1% to 20%, well within experimental expectations. ${\chi }_{\mathrm{Q}}/2\pi =5$ MHz, and the drive power $a_0$ is such that $|{\alpha }_{\mathrm{O}}{|}^2=9$.

Figure 5

$1-F_{01}$ as a function of the cavity occupation prior to detection and imperfect reset (horizontal axis) and the number of photons removed from the cavity by the photon detector (color and line style). The initial cavity states considered are maximally distinguishable, so this represents the worst-case scenario.

Figure 6

(a) Cavity occupation for even-parity qubit states with Jaynes-Cummings cavity-qubit coupling. (b) Bright- and dark-count rates and measurement contrast for the cavity photon numbers of (a), assuming the even state is from the right band (worst case).