Efficient state estimation on quantum processors

TL;DR

This work addresses the challenge of scalable quantum-state tomography for multi-qubit processors by introducing two entanglement-free, post-processing-free approaches based on polarization identities. The first builds a five-basis scheme compatible with LOCC to yield an informationally complete reconstruction, while the second constructs a minimal set of $2n+1$ tensor-product bases requiring only local gates for measurement. Both provide explicit analytic reconstructions without solving large inverse problems, and their scalability is validated through IBM QPU experiments up to $n=12$ qubits and remote entangled trapped-ion experiments. The results demonstrate high fidelities and reveal trade-offs between shot resources and entangling-gate usage, offering practical tomography options for benchmarking large quantum processors and distributed quantum computing.

Abstract

We present two scalable and entanglement-free methods for estimating the collective state of an n-qubit quantum computer. The first method consists of a fixed set of five quantum circuits-regardless of the number of qubits-that avoid the use of entanglement as a measurement resource, relying instead on classical communication between selected pairs of qubits. The second method requires only 2n+1 circuits, each of which applies a single local gate to one of the n qubits during the measurement stage. Unlike traditional estimation methods, our approaches do not require any costly post-processing procedure to estimate a quantum state, enabling scalability to relatively large system sizes. We experimentally compare both methods on freely available IBM quantum processors, and observe how the state estimation varies with increasing number of qubits and shots. We further validated our results by estimating the 4-qubit entangled state of two remote ion-trap quantum processors, demonstrating that the optimized 2n+1 tomographic scheme achieves estimates consistent with standard methods while using exponentially fewer measurements.

Figures (14)

• Figure 1: Schematic representation of the key ingredients addressed by our proposal. Some highly desirable properties of a tomographic method include an explicit reconstruction formula scott2006tightgoyeneche2015fivevargas2024nearjames2001measurement, hardware-efficient gates gross2010quantumPereira_2022guo2024quantum, a scalable number of measurements goyeneche2015fivevargas2024nearPereira_2022, low post-processing cost to reconstruct a positive semidefinite operator kaznady2008quantumgoyeneche2015five, and the absence of entanglement in the measurement stage james2001measurementcramer2010efficientPereira_2022. While existing methods typically exhibit only a subset of these desirable features, our approach stands out as a powerful and comprehensive method combining analytical simplicity with experimental scalability. This comprehensive optimization represents a significant advance in multi-qubit quantum state tomography, as demonstrated by the experimental results presented in Sections \ref{['sec:IBM']} and \ref{['sec:Oxford']}.
• Figure 2: Graphs $\mathcal{G}^{ent}_{2}$ and $\mathcal{G}^{ent}_{2}$ for a $2$-qubit system associated to (a) entangled and (b) separable measurements, respectively. Note that crossing edges ($\slash$ and \\) represent entangled measurement states, whereas parallel edges (--- and $|$) represent separable measurement states. In general, edges in the graph $\mathcal{G}$ correspond to separable measurement states if and only if they are also edges of the hypercubic graph.
• Figure 3: (Color online) Degree $3$ hypercubic graph with a hamiltonian cycle highlighted
• Figure 4: Classical-quantum circuit implementing bases $\tilde{\mathfrak{B}}_k$ with $k=1,\dots,4$. Horizontal single and double lines correspond to quantum and classical registers, respectively. Vertical single lines are always connected to quantum operations applied conditionally on the values stored in classical registers, with open circle corresponding to conditioned on $0$, filled circle to conditioned on $1$ and we use a standard notation for the classical OR gate. Vertical double line corresponds to operation applied to the classical register. Single square corresponds to set operation -- if the quantum gate has been applied, the corresponding classical register is set to 1, but not reverted. Double square corresponds to overwrite -- result of the corresponding measurement is written over whatever value the corresponding classical register had before.
• Figure 5: Quantum circuits for the generation of the 7 separable measurement basis for a $3$-qubit system.

Theorems & Definitions (9)

• Proposition 1
• Lemma 1
• proof
• Proposition 1
• Lemma 2
• proof
• proof
• Definition 1
• Definition 2