Bounding many-body properties under partial information and finite measurement statistics

TL;DR

This work tackles the challenge of obtaining rigorous, quantitative bounds on many-body quantum properties using partial information and finite-shot data. It combines measurement data with semidefinite-programming (SDP) relaxations based on moment matrices to derive probabilistic lower and upper bounds for linear and nonlinear quantities, such as purity, under physically motivated constraints (e.g., ground states, steady states, symmetries). The approach is scalable: by recasting the problem in terms of moments, enforcing PSD moment matrices, and incorporating linear system guarantees, the method achieves polynomial-time certification while delivering bounds with a prescribed confidence level. Numerically, the authors demonstrate tight bounds on open-system steady-state currents, ground-state energies, entanglement measures, and even a 50-qubit model, illustrating substantial accuracy gains when combining measurements with SDP relaxations and highlighting potential for real-world quantum-platform certification and validation.

Abstract

Calculating bounds of properties of many-body quantum systems is of paramount importance, since they guide our understanding of emergent quantum phenomena and complement the insights obtained from estimation methods. Recent semidefinite programming approaches enable probabilistic bounds from finite-shot measurements of easily accessible, yet informationally incomplete, observables. Here we render these methods scalable in the number of qubits by instead utilizing moment-matrix relaxations. After introducing the general formalism, we show how the approach can be adapted with specific knowledge of the system, such as it being the ground state of a given Hamiltonian, possessing specific symmetries or being the steady state of a given Lindbladian. Our approach defines a scalable real-world certification scheme leveraging semidefinite programming relaxations and experimental estimations which, unavoidably, contain shot noise.

Figures (7)

• Figure 1: A visualization demonstrating the comparison between bounds obtained from an SDP relaxation, from measurement data, and from the combined set. Note how the combined set has tighter bounds than either set individually.
• Figure 2: A pictorial representation of the open system described by Eq. \ref{['eqn:lindblad']}.
• Figure 3: Plot showing how the bounds on the heat current change versus the total number of shots $N_{\rm tot}$. Each strategy is identified by a different colour, and for a given strategy the upper curve represents the upper bound, while the lower curve represents the lower bound. All measurements strategies here are without Z measurements. "Measure" refers to measuring the first 100 Pauli strings that appear in the successive generation of Lindbladian constraints, "SDP" refers to the SDP relaxation without measurements (formally, Eq. \ref{['sdp-notation']} taking $K=0$), and "SDP & Measure" refers to the combination of both. The dashed lines represent the bound in the limit of infinite shots. Bounds involving measurements are valid under a confidence level of $99.7\%$. Each point is from $50$ repeats. Our SDP here uses a moment matrix of size $150\times150$, as well as $2000$ linear constraints from the Lindbladian. The open system here is a $3\times3$ grid of qubits with Lindbladian given by Eq. \ref{['eqn:lindblad']}, with parameters $\gamma_c=0.011$, $\gamma_h=0.001$, $T_h=1.0$, $T_c=0.1$, and $J=h=1$. Notably, the combined strategy (SDP & Measure) is the only one able to certify the positivity of the heat current, as the lower bound is greater than zero. Overall, its results are substantially more informative than the ones obtained via the other two strategies.
• Figure 4: Plot showing how the lower bound on the ground-state energy changes versus the total number of shots. Here "SDP & Measure (a)" refers to measuring the 100 Pauli strings that appear most frequently in the SDP, "SDP & Measure (b)" refers to measuring all second-order Pauli strings, "SDP & Measure (c)" refers to measuring only the Pauli strings that appear in the objective, whilst "SDP" refers to the SDP relaxation of the problem without any measurement data. The dashed lines represent the bound in the limit of infinite shots. Sets (b) and (c) have the same bound at infinite shots - the true ground state. All results here are to a confidence level of $99.7\%$. Each point is the mean from 50 repeats. Our SDP here uses a moment matrix of size $150\times150$. The system here is a $3\times3$ grid of qubits with Hamiltonian given by Eq. \ref{['eqn:hamil']} for $J=h=1$.
• Figure 5: Plot showing how the lower bound on the purity of the ground state changes versus the total number of shots. The known optimum is $1$ as the ground state is pure. Here "Measure" refers to measuring all second-order Pauli strings, "SDP" refers to the SDP relaxation without measurements, and "SDP & Measure" refers to the combination of both. The dashed lines represent the bound in the limit of infinite shots. All results here are to a confidence level of $99.7\%$. Each point is the mean from 50 repeats. Our SDP here uses a moment matrix of size $150\times150$. The system is a $3\times3$ grid of qubits with Hamiltonian given by Eq. \ref{['eqn:hamil']} for $J=h=1$.