Universal monitored dynamics in multimode bosonic systems

Abstract

We propose a route to study monitored many-body dynamics in multimode bosonic systems using circuit quantum electrodynamics. In this experimental setting, we construct several bosonic models comprising brickwork circuits built from beam-splitter gates, local parity measurements, and optional on-site Hubbard interactions, and diagnose their monitored dynamics via ancilla purification and a learnability-based probe. Under parity measurements, generic gate sets exhibit behavior that is largely consistent with a conventional measurement-induced phase transition, while a special class of beam-splitter circuits shows an apparent critical-like high-measurement regime in which purification times scale linearly with system size. We show that for realistic noise, gate, and measurement rates, these signatures are observable with near-term circuit QED hardware.

Figures (4)

• Figure 1: Multimode bosonic analog of a brick-layer circuit for probing monitored dynamics. The resonant modes of low-loss superconducting cavities act as harmonic oscillators encoding bosonic degrees of freedom. A reference qubit (dark blue Bloch sphere) is prepared in a globally entangled state with the modes and serves as a probe of the MIPT through its purification dynamics. Each layer applies random beam-splitter gates (light blue), optional on-site Hubbard-like gates (green), and probabilistic mid-circuit measurements (squares with arrows) of parity or photon number.
• Figure 2: Monitored dynamics with parity measurements. Ancilla entanglement entropy $S_R$ for circuits composed of (a) both BSFP and on-site Hubbard gates of strength $U=2$ and (b) those composed of purely BSFP gates ($U=0$) for systems of size $L=4-16$, averaged over 10000 independent realizations. For the $U=2$ case, both the clear crossing of the $S_R(p,t=2L)$ curves at $p_c \approx 0.3$ and the collapse of $S_R(p_c \approx 0.3, t/L)$ (inset) indicate the presence of a phase transition. In contrast, the $U=0$ data shows no clear signature of a standard MIPT, with the ancilla not purifying at any value of $p$ at these time scales. In addition, the collapse of $S_R(p,t/L)$ curves for the presented times for $p \gtrsim 0.3$ for the largest numerically-accessible system sizes ($L=12-16$, inset) indicates that at higher measurement rates, the system is in a phase with a purification time that scales with system size. (c) Comparison of $S_R(p=1,t)$ between the $U=2$ and $U=0$ cases. Turning on the on-site Hubbard gates takes the system from a phase in which the purification time scales with system size ($U=0$) to a phase with an $\mathcal{O} (1)$ purification time ($U=2)$. Learnability (d) Simulated decoder accuracy $A(p)$ for monitored dynamics with BSFP and on-site Hubbard gates ($U=2$). The accuracy $A(p)$ shows a similar transition to the ancilla entanglement entropy $S_R$ in (a) and may be used as an alternative post-selection free order parameter. Inset: protocol for detecting the learnability transition. A monitored circuit is run in the lab using an initial state of either $\ket{\psi_0}$ or $\ket{\psi_1}$, where $\braket{\psi_0 | \psi_1}=0$. A classical decoder then outputs a prediction of the initial state given the measurement record $\vec{m}$, the gates, and knowledge of $\ket{\psi_0}, \ket{\psi_1}$. Summary table (e) Summary of all numerical results. For non-BSFP gates with any measurement type, the data is consistent with a standard MIPT. In contrast, the purely BSFP gates are consistent with either a critical phase at high $p$ (parity measurements) or a purified phase at any $p$ (density measurements); "None" refers to the lack of an MIPT with $z=1$.
• Figure 3: Linear cavity array architecture (a) Schematic of a 1D $\lambda/4$ cavity array coupled via tuneable couplers. Each cavity is also interfaced with a chip hosting a transmon and readout cavity. Cascaded RAQM architecture (b) Multimode cQED setup of a cascaded random access quantum memory (RAQM) architecture. A superconducting storage cavity coupled to a buffer cavity via a tunable coupler; the buffer cavity is interfaced with a readout resonator through a transmon. Beam-splitter gate(c) Cavity modes modeled as harmonic oscillators are coupled via a tunable coupler (ex. SQUID, LINC, or SNAIL) (dashed box). Using a SNAIL we realize a 250 ns beam-splitter gate creating an entangled $\frac{1}{\sqrt2}(\ket{10}+\ket{01})$ state between the modes, and 500 ns SWAP gate. SNAP gate (d) SNAP operations, used to realize on-site Hubbard gates, apply geometric photon-number-dependent phases by driving the transmon qubit in a closed trajectory (shaded red) around the Bloch sphere in each photon number subspace. Parity measurement (e) Ramsey-based nondemolition photon-number parity readout using dispersive coupling, where two $\pi/2$-pulses are separated by some idle time $T=\pi/\chi$. State preparation (f) Pulse sequence for initializing a checkerboard-like state in the storage cavity entangled with a passive reference mode, using $\pi_{ge}/\pi_{ef}$ pulses, $f0g1$ sidebands, and coupler-mediated photon shuttling.
• Figure 4: Effects of noise in monitored bosonic circuits.(a) Averaged entropy curves for a model containing both BSFP and on-site Hubbard gates with 8 scrambling and 8 monitored layers. Results are shown for ideal gates (orange) and for circuits with all noise sources active (blue). Inset: the same curves after subtracting the residual entropy (RE) from the full loss curve. (b) Breakdown of residual entropies generated at $p=1$ (for a worst case estimate), obtained from simulations in which a single noise source is active at a time.