Quantum simulation in the Heisenberg picture via vectorization

TL;DR

This work presents a general framework for simulating quantum systems in the Heisenberg picture on quantum hardware based on the vectorization map, and demonstrates this by proposing implementations of the framework for a 2D problem on digital and analog quantum simulators, taking into account device connectivity constraints.

Abstract

We present a general framework for simulating quantum systems in the Heisenberg picture on quantum hardware. Based on the vectorization map, our framework fully exploits the mapping between operators and quantum states, allowing any task defined on Heisenberg operators to be mapped to standard Schrödinger-picture tasks that are naturally accessible via quantum computers and simulators. This yields new or improved protocols for tasks such as operator sampling, the computation of OTOCs/superoperator expectation values and their higher order moments, two-point correlators, and operator stabilizer and entanglement entropies. Our approach is also amenable to implementation, as it inherits the structure and resource requirements of the (forward and time-reversed) Schrödinger-picture quantum simulation problem. We demonstrate this by proposing implementations of our framework for a 2D problem on digital and analog quantum simulators, taking into account device connectivity constraints.

Figures (11)

• Figure 1: Schematic summarizing our algorithmic framework. Via the circuit schematic on the left, samples of the $2n$-qubit state $\mathinner{\!\left.\left|{O(t)}\right\rangle\right\rangle}_\mathcal{Q}$ are prepared via an initialize-and-evolve protocol, illustrated for the case of initializing and ending in the Pauli basis $\mathcal{P}$. Subsequently, by extracting information from it via measurements and other subroutines such as amplitude, fidelity, or trace estimation, a plethora of tasks in the Heisenberg picture can be accomplished naturally.
• Figure 2: Schematic of $\mathcal{H}'$ for $n=4$. The labeled $4$-qubit subsystem (mapping to a $2$-site operator) is enclosed in gray.
• Figure 3: Operator growth in the Heisenberg picture. Dynamics of initially localized operator on a 2D lattice; a single Pauli term of $O(t)$ is visualized. Operator geometrical properties -- such as its surface area, volume, etc. -- are functions of the local occupancies of each of the three 'flavors' of Pauli operators $\{X, Y, Z\}$, and are therefore expectation values of superoperators diagonal in the Pauli basis, $\langle \mathcal{D} \rangle_{O(t)}$.
• Figure 4: Circuit for computing two-point correlators. Exploiting unitality allows controlled time-evolution to be avoided. Measurement of the ancilla qubit in the $\{\ket{\pm}\}$ basis yields $c(O ,O')$.
• Figure 5: Implementation of our framework in a square-lattice device. (a) Initial embedding of the $2n = 18$ qubits of $\mathinner{\!\left.\left|{O}\right\rangle\right\rangle}$ onto a square grid. Red (blue) qubits occupy one half of the Hilbert space $\mathcal{H}_L$ ($\mathcal{H}_R$). Each red-blue qubit pair (delimited by gray capsules) encodes a single operator located at spatial position $i$. The effective propagator $e^{-iHt} \otimes e^{iH^\mathsf{T}t}$, once Trotterized, contains entangling gates that act between nearest and next-nearest-neighbors in a square grid. This is represented as edges between black nodes (each representing a local operator) of the graph at the right. Within a single Trotter step, there will be 24 entangling gates, representing by the 24 (curved) edges in this graph. (b) and (c) shows the transversal Hadamard and CNOT gates carrying out the basis transformation $R_{\mathcal{P} \rightarrow \mathcal{C}}$.

Theorems & Definitions (31)

• Theorem 1: $\alpha$-stabilizer purities
• Theorem 2: Linear entropy
• Theorem 3: Commuting superoperators
• Corollary 1
• Corollary 2: Diagonal superoperators
• Corollary 3: Estimating all diagonal OTOCs
• Theorem 4: Two-point correlators, commuting superoperators
• Corollary 4: Diagonal two-point correlators
• Theorem 5: Operator sampling with $n$ qubits
• Corollary 5: $n$-qubit analogues of Theorems \ref{['theorem:simul_otoc']} and \ref{['theorem:simul_2pc']}