Postselection-Free Entanglement Dynamics via Spacetime Duality

The dynamics of entanglement in “hybrid” nonunitary circuits (for example, involving both unitary gates and quantum measurements) has recently become an object of intense study. A major hurdle toward experimentally realizing this physics is the need to apply postselection on random measurement outcomes in order to repeatedly prepare a given output state, resulting in an exponential overhead. We propose a method to sidestep this issue in a wide class of nonunitary circuits by taking advantage of spacetime duality. This method maps the purification dynamics of a mixed state under nonunitary evolution onto a particular correlation function in an associated unitary circuit. This translates to an operational protocol which could be straightforwardly implemented on a digital quantum simulator. We discuss the signatures of different entanglement phases, and demonstrate examples via numerical simulations. With minor modifications, the proposed protocol allows measurement of the purity of arbitrary subsystems, which could shed light on the properties of the quantum error correcting code formed by the mixed phase in this class of hybrid dynamics.

Figure 1

Spacetime duality. By swapping the spatial and temporal axes, a unitary gate $U$ maps onto another, generally nonunitary gate $\tilde{U}$.

Figure 2

(a) Purification dynamics: a fully mixed state ${\rho }_{\mathrm{in}}\propto \mathbb{1}$ (gray lines, bottom) is evolved by a hybrid circuit $M$, yielding a state ${\rho }_{\mathrm{out}}\propto M{M}^{†}$ (purple lines, top). (b) If $M={\tilde{U}}_M$ with $U_M$ unitary, the purification process is spacetime dual to a unitary evolution. ${\rho }_{\mathrm{out}}$ lives on a timelike slice of the circuit (purple legs, right). (c) Purity of the output state, $\mathrm{Tr}({\rho }_{\mathrm{out}}^2)$. (d) Same tensor network viewed as a correlation function in a unitary circuit. The arrow of time $\tilde{t}$ ($t$) denotes hybrid (unitary) evolution. (e) Protocol for measuring the purity of ${\rho }_{\mathrm{out}}$, sketched for $T=3$. Tensor network legs are color coded as in (a)–(d). Qubits $\pm 1$ are repeatedly initialized in the Bell state $P=|B^{+}⟩⟨B^{+}|$ (upward purple arcs) and measured in the Bell basis (downward purple arcs); the protocol succeeds if all $T$ Bell measurements yield $|B^{+}⟩$. (f) The fraction of runs that are successful up to time $T$, $N_{+}(T)/N_{\mathrm{tot}}$, yields the purity of ${\rho }_{\mathrm{out}}$ on $\tilde{L}=2T$ qubits.

Figure 3

(a) Summary of the Clifford circuit model: probabilities of $\mathbb{1}$, $\mathsf{i}\mathsf{S}\mathsf{W}\mathsf{A}\mathsf{P}$ and $\mathsf{S}\mathsf{W}\mathsf{A}\mathsf{P}$ gates. (b) Schematic of purification phase diagram of the dualized circuit vs $p$, $J$. (c) Entropy density of hybrid circuit output state ${\rho }_{\mathrm{out}}$ vs $p$, $J$ (Clifford simulations on $\tilde{L}\le 4096$ qubits).

Figure 4

Critical phase of the noninteracting ($J=0$) Clifford circuit model. (a) Allowed gates, $\mathbb{1}$ (blue tile) and $\mathsf{S}\mathsf{W}\mathsf{A}\mathsf{P}$ (red tile). (b) A realization of the circuit for $T=5$. The purification dynamics proceeds left to right; the input qubit worldlines (${\rho }_{\mathrm{in}}$, left) are fully mixed; the output state (${\rho }_{\mathrm{out}}$, right) contains Bell pairs (arcs) and fully mixed qubits ($\mathbb{1}$ symbols). (c) Coarse-grained view of the purification dynamics ($T\gg 1$). Only qubit worldlines that enter within a distance $\xi \sim T^{1/2}$ of ${\rho }_{\mathrm{out}}$ (e.g., green shaded parabola) are likely to diffuse to ${\rho }_{\mathrm{out}}$ and contribute entropy.