Scalable Experimental Bounds for Entangled Quantum State Fidelities

TL;DR

This work tackles the scalability bottleneck of verifying entanglement on NISQ devices by deriving and experimentally validating scalable fidelity lower bounds for Dicke states up to $N=10$ and GHZ states up to $N=20$ using symmetry-based operators that require only a few measurement settings. The authors implement efficient Dicke-state preparation circuits with linear-depth and discuss an even more scalable approach via approximate Dicke states (Product States and Even/Odd Hamming-weight states), demonstrating that meaningful lower bounds can be achieved on Quantinuum hardware with modest shot budgets. The results show bounds that match or surpass exact fidelities reported on much smaller superconducting devices, and they provide practical guidance for benchmarking entanglement as quantum hardware scales. Collectively, the work offers a concrete path toward scalable entanglement benchmarking on next-generation NISQ devices, balancing circuit resources, measurement overhead, and bound tightness through both exact and approximate state preparations.

Abstract

Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is important for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as $O(3^N)$ for $N$-qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma and others [PhysRevA.74.052302] established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke States and GHZ States. These bounds must still be tight enough for larger states to provide reasonable estimations on NISQ devices. For the first time and more than 15 years after the theoretical introduction, we report meaningful lower bounds for the state preparation fidelity of all Dicke States up to $N=10$ and all GHZ states up to $N=20$ on Quantinuum H1 ion-trap systems using efficient implementations of recently proposed scalable circuits for these states. Our achieved lower bounds match or exceed previously reported exact fidelities on superconducting systems for much smaller states. Furthermore, we provide evidence that for large Dicke States $D^N_{N/2}$, we may resort to a GHZ-based approximate state preparation to achieve better fidelity. This work provides a path forward to benchmarking entanglement as NISQ devices improve in size and quality.

Figures (9)

• Figure 1: Experimental fidelities of Dicke states $\ket{{D_{K}^{N}}}$ (by circuit complexity) on IBMQ devices based on full state tomography data aktar2022divide: (red) Exact fidelity computed from all $3^N$ tomography measurement settings, (green) Upper bound computed only from the $Z$-basis measurement data. (blue) We compute fidelity lower bounds from experimental $X$-, $Y$-, and $Z$-basis measurement setting subsets of their data, slightly improving on direct application of existing lower bound techniques somma2006lower$\textit{(orange)}$.
• Figure 2: Divide-and-conquer Dicke state $\ket{{D_{3}^{9}}}$ preparation allowing parallelizable Dicke state unitaries on $4$ and $5$ qubits: (1) We prepare a correctly weighted superposition of input Hamming weights $0\leq\ell\leq 3$ on the first register. The used $R_y$-rotations contain arguments with numerators (denominators) derived from (suffix-sums) of terms of the form $\tbinom{5}{3-\ell}\tbinom{4}{\ell}$, the number of distributions of $\ell$ & $3-\ell$ Ones across 4 & 5 qubits. (2&3) The first register is correctly entangled with the second register, and bit-flipping $X$-gates are applied to reduce the number of CNOTs in the following. (4&5) Parallel Dicke state unitaries $U_{1,4}^4$ and $U_{2,5}^5$ prepare the Dicke state $\ket{{D_{6}^{9}}}$, followed by bit-flipping $X$-gates to get $\ket{{D_{3}^{9}}}$.
• Figure 3: Upper and lower bounds on the quantum fidelity $\bra{{D_{K}^{N}}}\rho\ket{{D_{K}^{N}}}$ for Dicke States $\ket{{D_{K}^{N}}}$ on the Quantinuum H1-2 Processor (top) and its H1-2E Simulator (bottom), including 68% confidence intervals. Dicke states are sorted along the $x$-axis according to the number of CNOTs in their preparation circuits. Compared to the simulator, the Quantinuum H1-2 Processor demonstrates better upper and lower bounds on the Dicke state preparation fidelity. The bump in fidelity on Quantinuum H1-2 for Dicke states $\ket{{D_{1}^{8}}}$--$\ket{{D_{4}^{8}}}$ and $\ket{{D_{1}^{9}}}$--$\ket{{D_{2}^{9}}}$ may be due to these experiments taking place right after a recalibration downtime of the QPU.
• Figure 4: Evolving bounds and confidence intervals on Dicke state fidelity $\bra{{D_{K}^{N}}}\rho\ket{{D_{K}^{N}}}$ for $\ket{{D_{4}^{8}}}$ and $\ket{{D_{4}^{10}}}$ on Quantinuum H1-2 with increasing shot count. Experiments were run for $280$ and $840$ shots for $\ket{{D_{4}^{8}}}$ and $\ket{{D_{4}^{10}}}$, respectively, to achieve comparable confidence intervals. Plots are scaled (1:3) accordingly.
• Figure 5: Preparation of a ($N=9$)-qubit GHZ state $\ket{{G_{9}}}$: (left) Schematics, (center) Linear-depth circuit on LNN connectivity, (right) Logarithmic-depth circuit on full connectivity. For the linear depth GHZ circuit, all CNOT gates are executed consecutively and generate a linear time $O(N)$ complexity. For the logarithmic depth GHZ circuit, the gates leading to (1),(2),(3) & (4), respectively, can be executed in parallel and generate a logarithmic $O(\log N)$ complexity.