Emergent quantum mechanics at the boundary of a local classical lattice model

We formulate a model in which quantum mechanics emerges from classical mechanics. Given a local Hamiltonian $H$ acting on $n$ qubits, we define a local classical model with an additional spatial dimension whose boundary dynamics is approximately—but to arbitrary precision—described by Schrödinger's equation and $H$. The bulk consists of a lattice of classical bits that propagate towards the boundary through a circuit of stochastic matrices. The bits reaching the boundary are governed by a probability distribution whose deviation from the uniform distribution can be interpreted as the quantum-mechanical wave function. Bell nonlocality is achieved because information can move through the bulk much faster than the boundary speed of light. We analytically estimate how much the model deviates from quantum mechanics, and we validate these estimates using computer simulations.

Figure 1

Square lattice picture of our model (with $S=5$ and $n=4$), which exhibits EmQM on a one-dimensional boundary (circled in red) of a two-dimensional bulk. Classical bits $a_{s,x}$ (red dots) propagate forward through a slowly time-evolving circuit of stochastic matrices $M_{s,x}$ (yellow), and classical bits $b_{s,x,\gamma }$ propagate backwards through a time-independent circuit of permutation matrices $Q_{s,x}^T$. The emergent Schrödinger equation describes the time evolution of the probability distribution $P\left(a_S\right)$ governing the boundary bits. The back-propagating bits determine how the stochastic matrices are updated in each time step. See Sec. 3a for an overview.

Figure 2

(a, b) To validate our error estimates, we plot the deviation $\varepsilon \left(t\right)=\left|\right|\mathrm{\Psi }\left(t\right)-{\mathrm{\Psi }}_{\text{QM}}\left(t\right)\left|\right|$ vs time $t$ between the quantum mechanics (QM) wave function ${\mathrm{\Psi }}_{\text{QM}}$ and emergent QM (EmQM) wave function $\mathrm{\Psi }$ [Eq. (51)] for (a) $n=4$ and (b) $n=6$ qubits using three different random initializations (colored lines) for each choice of the control parameter ${\epsilon }_0$ [see Eq. (79)]. As ${\epsilon }_0$ decreases, the EmQM model becomes increasingly more accurate, roughly agreeing with QM out to time $t\sim {\epsilon }_0^{-1}$. Dashed black lines show the estimated deviation $\varepsilon \left(t\right)\sim \sqrt{{\epsilon }_{0}t+3{\left({\epsilon }_{0}t\right)}^2}$ [Eq. (78)] for the three different values of ${\epsilon }_0$, which match the simulated data remarkably well. (c, d) To demonstrate that the EmQM model is reproducing nontrivial dynamics, we also plot the 0110 (and 010110) component of the $n=4$ (and $n=6$) wave functions vs time for QM and the EmQM model with ${\epsilon }_0=0.02$ (and ${\epsilon }_0=0.05$), for which $S=800$ (and $S=1280$) [in accordance with Eq. (79)].

Figure 3

Deviation $\varepsilon \left(t\right)=\left|\right|\mathrm{\Psi }\left(t\right)-{\mathrm{\Psi }}_{\text{QM}}\left(t\right)\left|\right|$ from quantum mechanics (QM) vs time $t$. The deviation is dominated by statistical fluctuations ${\varepsilon }_{\text{stat}}\left(t\right)$ until the change in slope, after which the deviation is dominated by other contributions. Blue, yellow, and red lines show dominant contributions from ${\varepsilon }_{\text{S}}\left(t\right), {\varepsilon }_{\text{t}}\left(t\right)$, and ${\varepsilon }_{\text{S}}\left(m\right)$, respectively, defined in the legend (with ${\varepsilon }_{\text{stat}}\left(t\right)\sim {10}^{-2}\sqrt{{\epsilon }_{0}t}$ and ${\varepsilon }_{\text{S}}\left(t\right)\sim {\varepsilon }_{\text{t}}\left(t\right)\sim {\varepsilon }_{\text{m}}\left(t\right)\sim {\epsilon }_{0}t$ whenever unspecified). Simulations are for $n=4$ qubits and ${\epsilon }_0=0.05$ with parameters $S, {\delta }_{\text{t}}, m_0$, and ${\mathrm{\Delta }}_{\text{m}}$ chosen to target the previously mentioned contributions [using Eqs. (63), (65), and (70)]. The dashed lines plot the expected deviations $\varepsilon \left(t\right)$ using Eq. (76), which agree remarkably well with the simulation data (colored lines).

Figure 4

Deviation $\left|\right|\mathrm{\Psi }\left(t\right)-{\mathrm{\Psi }}_{\text{QM}}\left(t\right)\left|\right|$ vs time $t$ for $n=4$ qubits with ${\epsilon }_0=1$, showing agreement between the EmQM model (gray), approximate algorithm with ${\epsilon }_{\text{j}}=0.02$ (red), and estimated $\varepsilon \left(t\right)$ from Eq. (78) (dashed black). For the EmQM model and approximate algorithm, we average over 1000 random realizations and plot the mean (thick lines) along with the mean plus or minus the standard deviation (thin lines) to demonstrate that both the mean and statistical fluctuations agree well. Error bars denote one standard deviation of statistical error resulting from the finite number of 1000 samples.

Figure 5

Deviation $\left|\right|\mathrm{\Psi }\left(t\right)-{\mathrm{\Psi }}_{\text{QM}}\left(t\right)\left|\right|$ (red) vs time $t$ calculated using the approximate algorithm with a large ${\varepsilon }_{\text{jump}}=10$ so that we can study errors due to the approximate simulation algorithm. Simulations use $n=4$ qubits and three different random initializations with $\varepsilon =0.01$ [in Eq. (79)]. In all simulations in this work, ${\mathrm{\Delta }}_{\text{jump}}$ is chosen to be as large as possible without exceeding ${\mathrm{\Delta }}_{\text{jump}}^{\text{max}}$. ${\mathrm{\Delta }}_{\text{jump}}$ is sometimes limited (shown by blue dots) when the time step is very small (due to the logarithmic time axis). By inserting the actual time-dependent ${\mathrm{\Delta }}_{\text{jump}}\left(t\right)$ into Eq. (D4), our estimate $\tilde{\varepsilon }\left(t\right)$ [(D7), dashed black line] for the deviation (under the influence of the approximation) matches the ${\varepsilon }_{\text{jump}}\left(t\right)$-dominated simulation data (red) remarkably well. In contrast, the dashed gray curve shows the expected deviation $\varepsilon \left(t\right)$ of the EmQM model without approximation, which is significantly smaller when $0.01\lesssim t\lesssim 10$. This validates Eq. (D7).