Minimum measurements quantum protocol for band structure calculation

TL;DR

The paper addresses the measurement overhead in quantum simulations of band structures by deriving a symmetry-driven constant three-round protocol that eliminates qubit-count scaling of measurement settings. It employs a reciprocal orbital qubit mapping to convert tight-binding Hamiltonians into a Pauli-based qubit form and uses three measurement bases to extract all required correlators for the VQD objective. A reconstruction rule and zero-amplitude compression enable complete cost-function evaluation with constant overhead, demonstrated on CuO2 and bilayer graphene benchmarks and validated up to 14-qubit correlator estimates. The work shows that, for symmetry-rich Hamiltonians, measurement costs can be made size-invariant, offering a pathway toward scalable quantum advantage on near-term devices.

Abstract

Protocols for quantum measurement are an essential part of quantum computing. Measurements are no longer confined to the final step of computation but are increasingly embedded within quantum circuits as integral components of noise-resilient algorithms. However, each observable typically requires a distinct measurement basis, often demanding a different circuit configuration. As the number of such configurations typically grows with the number of qubits, different measurement configurations constitute a major bottleneck. Focusing on electronic structure calculations in crystalline systems, we propose a measurement protocol that maximally reduces the number of measurement settings to just three, independent of the number of qubits. This makes it one of the few known protocols that do not scale with qubit number. In particular, we derive the measurement protocol from the symmetries of tight-binding (TB) Hamiltonians and implement it within the Variational Quantum Deflation (VQD) algorithm. We demonstrate its performance on two systems, namely a two-dimensional CuO$_2$ square lattice (3 qubits) and bilayer graphene (4 qubits). The protocol can be generalized to more complex many-body Hamiltonians with high symmetry, providing a potential path toward future demonstrations of quantum advantage.

Figures (6)

• Figure 1: Three measurement circuits $\mathcal{M}_{Z}$, $\mathcal{M}_{XX}$, $\mathcal{M}_{XY}$ for cost function, Eg. \ref{['eq.cost_fun']}, estimation. (a) The measurement circuit for estimating the probabilities $\abs{a_{j}}^{2},\, j=0,1,\ldots,N-1$. (b) The measurement circuit for estimating all $\langle\hat{X}_{j}\hat{X}_{l}\rangle$ terms for $j=0,1,\ldots N-2, l > j$. (c) The measurement circuit for estimating $\langle\hat{X}_{j}\hat{Y}_l\rangle$ for all qubit indices $j, l$ with different parity. The circuit depicted here is the special case when $N$ is even number. If $N$ is odd number, the sequence ends with the Hadamard gate on the last qubit. (d) Measurement gates used to rotate qubits into the common eigenbasis prior to measurement.
• Figure 2: Example of the measurement strategy for the 4-qubit model. Grey indicates zero amplitudes, while blue and purple denote the application of measurement gates that rotate the qubits into the common eigenbasis: $X$ and $Y$, respectively. (a) All possible situations for zero amplitude (grey color) events for $h = 1$. The corresponding Pauli correlators that contribute to the cost function, see Eq. \ref{['eq.cost_fun']}, are obtained directly from the measurements (green) and indirectly using the product rule formula \ref{['eq.prod_rule']} (red). (b) All possible situations for zero amplitude (grey color) events for $h = 2$. The corresponding Pauli correlators that contribute to the cost function, see Eq. \ref{['eq.cost_fun']}, are obtained directly from the measurements (green). There are no indirect Pauli correlators needed.
• Figure 3: Workflow of the constant measurement protocol. The QPU denotes the parts of the algorithm that are carried out by a quantum computer or simulator. The input parameters are the variational angles $\boldsymbol{\theta}_{i}$ and the model parameters $\varepsilon_{j}$, $\mathcal{H}(\boldsymbol{k})_{jl}$. $N_{\text{max}}$ denotes the maximum number of iterations. The symbol $h$ represents the number of zero amplitudes. The green shaded parts depict the classical post-processing.
• Figure 4: Total number of circuit executions (log scale) for the cost function estimation, comparing conventional and constant measurement protocols with $N_{\text{shots}} = 10^4, 10^5, 10^6$.
• Figure 5: Comparison of two models. (a) CuO$_2$ structure - unit cell with three atomic orbitals and the relevant hopping amplitudes. (b) The first Brillouin zone of CuO$_2$ and the high-symmetry path. (c) CuO$_2$ band structure (red crosses: VQD results; blue lines: exact diagonalization, three-qubit model). (d) Bilayer graphene structure - Monolayer top view and the first Brillouin zone with the high-symmetry path. (e) Bilayer graphene side view with relevant hopping amplitudes. (f) Bilayer graphene band structure (red crosses: VQD results; blue lines: exact diagonalization, four-qubit model).