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Practical characterization of quantum devices without tomography
Marcus P. da Silva, Olivier Landon-Cardinal, David Poulin
TL;DR
Quantum tomography becomes exponentially costly with system size, while experiments often need only targeted fidelity or model-identification information. The paper develops Monte Carlo certification and learning schemes using local measurements and product-state preparation, extending them to qubits and continuous variables. Fidelity estimation uses a system-size-independent number of measurement settings, though sampling and measurement costs retain important state- and setting-dependent boundaries.
Problem
Tomography requires experimental and post-processing resources that increase exponentially with the number of particles, although experiments often seek only targeted information such as fidelity to a target.
Method
The paper develops targeted certification and learning approaches using local measurements and product-state preparation, with Monte Carlo relevance sampling for fidelity estimation.
Results
The number of fidelity-estimation measurement settings is independent of system dimension and scales as O(1/ϵ^2).
Takeaways & Limitations
The schemes provide a practical route to characterizing qubits and continuous-variable systems without full tomography.
Takeaways & Limitations
For general pure states, compressed sensing may improve the proposed algorithm's performance, indicating that the presented sampling approach is not necessarily optimal.
Abstract
from arXiv · showhide
Quantum tomography is the main method used to assess the quality of quantum information processing devices, but its complexity presents a major obstacle for the characterization of even moderately large systems. The number of experimental settings required to extract complete information about a device grows exponentially with its size, and so does the running time for processing the data generated by these experiments. Part of the problem is that tomography generates much more information than is usually sought. Taking a more targeted approach, we develop schemes that enable (i) estimating the fidelity of an experiment to a theoretical ideal description, (ii) learning which description within a reduced subset best matches the experimental data. Both these approaches yield a significant reduction in resources compared to tomography. In particular, we demonstrate that fidelity can be estimated from a number of simple experimental settings that is independent of the system size, removing an important roadblock for the experimental study of larger quantum information processing units.
Statistical bound for Monte-Carlo estimation of the fidelity
The Monte Carlo fidelity estimator samples observables according to their relevance to the target state and provides rigorous finite-error bounds. Its measurement-setting count is system-size independent, while truncation controls costly low-relevance observables.
- Rigorous bounds quantify the error of the Monte Carlo fidelity estimate for n-qubit systems.
- The estimator samples indices from Pr(i)=ρ_i^2/d and averages independent estimates of σ_i/ρ_i.The sampled observables are experimentally measured on the state under test.
- The number of measurement settings is independent of system dimension and scales as O(1/ϵ^2).
- Sampling remains costly for general states, but local-observable expectation values can be computed in polynomial time for many tensor-network families, including MPS.
- For W and |t_n⟩ states, the smallest nonzero Pauli expectation scales as 1/n, requiring n^2 samples per observable for constant accuracy in the worst case.
- Discarding expectation values below the relevance threshold yields O(2^{n(1+ϵ)}) total measurements for qubits, at least a quadratic improvement over brute-force tomography.
Extension to continuous variables systems
The Monte Carlo fidelity method extends to continuous-variable systems by replacing discrete operator-basis sampling with phase-space sampling. Wigner functions provide a representation compatible with both probability sampling and experimentally accessible expectation estimates.
- The method adapts to continuous-variable systems by changing state parameterization and the sampling procedure.
- Phase-space quasiprobability distributions represent states and observables while preserving the fidelity-estimation framework.
- The relevance density is sampled over phase space, with regions of zero density excluded and standard rejection sampling available.
- Wigner functions satisfy the required representation conditions: they are self-dual, bounded in magnitude by 2, and experimentally estimable at chosen phase-space points.
- A cutoff can disregard regions with relevance density below c, after which the fidelity error is bounded.
Sampling from the relevance distribution
Directly sampling the relevance distribution is difficult because the operator space and the set of target-state expectation values can grow exponentially. Tensor-product operators enable recursive sampling instead.
- Sampling Pr(i) directly is inefficient because the operator space on n particles is exponentially large.
- Computing all target-state coefficients ρ_i, or even one coefficient, can itself be challenging.
- Choosing tensor products of single-particle operators, such as Pauli operators, permits recursive observable selection.
Sampling using conditional probabilities
For tensor-product operator bases, the relevance distribution factorizes through conditional probabilities. This converts global sampling into sequential local-observable choices based on previously selected observables.
- For a tensor-product observable, the relevance distribution can be written as conditional probabilities for successive particles.
- The sampler sequentially chooses each single-particle observable according to its conditional distribution.
- The conditional probabilities use two-copy expectation values, reflecting the squared relevance weights.
- For Pauli bases, the operator connecting duplicated particle pairs is the SWAP operator.
- The duplicated-observable construction corresponds to expectation values of tensor products of 2-local observables on ρ⊗ρ.
Bound on the complexity of sampling
Sampling from the relevance distribution can be efficient for states whose local-observable expectations are efficiently computable, including tensor-network and computationally tractable states. For generic states, the procedure can have worst-case exponential complexity, although compressed sensing may improve performance for general pure states.
- Efficient state families: For tensor-network states such as MPS, local-observable expectations are often computable in polynomial time, enabling efficient relevance-distribution sampling.For MPS specifically, the procedure can be simplified to sampling complexity linear in n; PEPS permit efficient heuristic expectation-value computation.
- Efficient state families: Computationally tractable states support efficient sampling because their computational-basis overlaps and measurement-outcome distributions can be computed or sampled efficiently.
- Generic complexity: For generic n-qubit state vectors, computing a single local-observable expectation can require O(2^2n) time, and sampling repeats this computation across particles.
- Generic complexity: Compressed sensing can recover low-rank density matrices from O(n 2n) expectation values, suggesting possible improvement over the proposed algorithm for general pure states.
Lieb-Robinson bound
The Lieb-Robinson bound makes local dynamics effectively causal: an observable’s evolution can be approximated using a Hamiltonian truncated to a nearby region, with errors decreasing exponentially with distance. This locality removes the general learning obstacles and supports stable numerical Hamiltonian estimates under finite statistical error.
- Local dynamics: A local Hamiltonian is a sum of bounded-norm terms acting on at most k neighboring particles, and operator evolution obeys ∂A(t)/∂t = i[H,A].
- Tensor-network representation: For an MPS state, Eq. (21) is represented by a tensor network whose contractions encode the relevance-distribution calculation.The figure uses colored tensors for Ω operators, selected Pauli operators, and copies of the MPS state and its bra.
- Lieb-Robinson bound: Under local Hamiltonian or Lindbladian evolution, evolving A with the full generator is approximated by evolution under the generator truncated to region R, with corrections decaying exponentially in d.Region M is a constant-thickness membrane surrounding R.
- Lieb-Robinson bound: Choosing d ≈ vt + log(cV/c′t) controls the truncation region while the Hamiltonian and surrounding membrane norms grow as d^D and d^(D−1), respectively.
- Hamiltonian learning: Short-time measurements produce linear constraints W_ij = it⟨[H,A_i]⟩_ρ, which become an invertible linear system for learning Hamiltonian coefficients.The coefficients are obtained by expanding H in an operator basis and inverting the resulting equation.
- Hamiltonian learning: For local Hamiltonians, locality removes the general approach’s caveats, and numerical experiments show well-behaved error scaling even with finite statistical error.The experiments use small qubit chains with random nearest-neighbor interactions and Moore-Penrose pseudoinversion.