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Variational Quantum Fidelity Estimation
M. Cerezo, Alexander Poremba, Lukasz Cincio, Patrick J. Coles
TL;DR
Computing fidelity matters for verifying and characterizing mixed quantum states, but direct classical computation can scale exponentially. The paper introduces VQFE, which uses truncated-fidelity bounds for low-rank fidelity estimation. The bounds can be refined with m, often outperform existing computable bounds, and can detect quantum phase transitions.
Problem
Direct classical fidelity computation may scale exponentially with density-matrix dimension, motivating efficient estimation for mixed states prepared on quantum computers.
Source: §1 · p. 1Method
Results
The authors report that their bounds typically outperform sub- and super-fidelities and are useful for detecting quantum phase transitions.
Source: §8 · p. 5Takeaways & Limitations
The fidelity spectrum is argued to provide information about state closeness beyond a single fidelity value.
Source: §1 · p. 2The effects on the bounds of reconstructing a non-positive T matrix with maximum-likelihood methods remain unanalyzed, and such procedures can bias estimated entanglement properties.
Source: §8 · p. 5
Abstract
from arXiv · showhide
Computing quantum state fidelity will be important to verify and characterize states prepared on a quantum computer. In this work, we propose novel lower and upper bounds for the fidelity $F(\rho,\sigma)$ based on the "truncated fidelity" $F(\rho_m, \sigma)$, which is evaluated for a state $\rho_m$ obtained by projecting $\rho$ onto its $m$-largest eigenvalues. Our bounds can be refined, i.e., they tighten monotonically with $m$. To compute our bounds, we introduce a hybrid quantum-classical algorithm, called Variational Quantum Fidelity Estimation, that involves three steps: (1) variationally diagonalize $\rho$, (2) compute matrix elements of $\sigma$ in the eigenbasis of $\rho$, and (3) combine these matrix elements to compute our bounds. Our algorithm is aimed at the case where $\sigma$ is arbitrary and $\rho$ is low rank, which we call low-rank fidelity estimation, and we prove that a classical algorithm cannot efficiently solve this problem. Finally, we demonstrate that our bounds can detect quantum phase transitions and are often tighter than previously known computable bounds for realistic situations.
1 Introduction
Fidelity is important for verifying and characterizing mixed quantum states, but its direct computation may scale exponentially. The paper targets low-rank fidelity estimation with refinable bounds and a hybrid quantum-classical algorithm.
1 Introduction
- Fidelity serves verification against a target state and characterization of subsystem behavior near phase transitions.The paper motivates these uses for mixed states prepared intentionally or through quantum-computer noise.Source: §1 · p. 1
- Classically estimating fidelity may scale exponentially with density-matrix dimension, while direct finite-copy quantum estimation is obstructed by non-integer powers of the states.The authors also note that deciding whether trace distance is large or small is QSZK-complete [10].Source: §1 · p. 1
- The authors define Low-Rank Fidelity Estimation for arbitrary σ and approximately low-rank ρ, and prove that classical algorithms cannot solve it efficiently under standard assumptions.They note low-rank states occur in condensed matter physics and data science.Source: §1 · p. 1
- VQFE bounds F(ρ, σ) using truncated fidelities for ρ_m, with bounds that tighten monotonically and equal fidelity when m = rank(ρ).ρ_m projects ρ onto the subspace of its m largest eigenvalues.Source: §1 · p. 1
- VQFE often outperforms the sub- and super-fidelity bounds and uses 2n + 1 qubits, compared with their 4n + 1 qubits for n-qubit states.The authors say those bounds generally loosen when both states have high rank.Source: §1 · p. 2
- The fidelity spectrum collects truncated fidelity bounds across m and is argued to reveal more about state closeness than fidelity alone.The authors compare this added information to that provided by the entanglement spectrum over a single entanglement measure.Source: §1 · p. 2
2 Truncated fidelity bounds
The truncated fidelity bounds use a state projected onto the subspace of ρ’s m largest eigenvalues. Their fidelity spectrum tightens as m grows and can be related to the truncation error for approximately low-rank states.
2 Truncated fidelity bounds
- The truncated fidelity bounds are lower and upper bounds on fidelity, using truncated states and the truncated generalized fidelity.The generalized fidelity is defined for sub-normalized states and reduces to ordinary fidelity if at least one state is normalized.Source: §2 · p. 2
- The fidelity spectrum collects these bounds across m; they tighten monotonically and equal fidelity when m = rank(ρ).Increasing m therefore refines the bounds to arbitrary tightness.Source: §2 · p. 2
- For m = rank_ϵ(ρ), the truncated-fidelity-bound looseness is at most √ϵ.The ϵ-rank is the smallest m for which the trace norm of ρ − ρ_m is at most ϵ; rank_0(ρ) equals rank(ρ).Source: §2 · p. 2
3 The VQFE algorithm
VQFE combines variational diagonalization, measurements of σ in ρ’s eigenbasis, and classical post-processing to bound fidelity. Certified bounds account for nonzero diagonalization cost and tighten with the truncation size.
3 The VQFE algorithm
- VQFE uses three stages: approximately diagonalize ρ, measure σ in ρ’s eigenbasis, then classically process the measurements into fidelity bounds.The bounds use truncated fidelity and can be refined as m increases.Source: §3 · pp. 2–3
- VQSD variationally minimizes a cost C measuring how far the transformed state is from diagonal; C = 0 gives exact diagonalization.For nonzero C, VQSD outputs approximate eigenvalues and eigenvectors of a state ρ′.Source: §3 · p. 3
- For fixed m, the measurement subroutine estimates m(m + 1)/2 matrix elements σij, adding only 2m + 1 new quantities when m increases by one.It prepares eigenvector superpositions and uses a Swap Test to estimate their inner products with σ.Source: §3 · p. 3
- The classical step forms the positive semidefinite matrix Tij = √rirjσij and diagonalizes it to compute the truncated-fidelity bounds.If shot noise makes T non-positive, a maximum-likelihood positive matrix can replace it.Source: §3 · p. 3
- When the diagonalization cost is appreciable, certified bounds account for the difference between ρ′ and ρ using cost-function or triangular-inequality bounds.The cost-function bounds use δ2, which is directly computable from experimental data; δ1 is useful when ρ is promised low rank.Source: §3 · p. 3
- Certified Bounds combine the tighter certified upper and lower bounds from the cost-function and triangular-inequality propositions.The triangular-inequality bounds can use fidelity-monotone distances, including the Bures angle and Bures distance.Source: §3 · p. 3
4 Implementations
Simulator implementations on three- and six-qubit tensor-product states show rapidly converging truncated-fidelity bounds and tighter certified bounds than SSFB.
4 Implementations
- With cost C ∼10−6, the simulator used random σ and tensor-product ρ states diagonalizable by a depth-one circuit ansatz.The examples used COBYLA in the VQSD optimization loop and considered n = 3 and n = 6 qubits.Source: §4 · p. 4
- As m increases, the truncated-fidelity bounds rapidly converge to F(ρ, σ) in both simulator examples.Because C was small, the authors treat these as bounds on F(ρ, σ).Source: §4 · p. 4
- In both examples, the Certified Bounds are significantly tighter than SSFB.The caption notes a remaining gap for the six-qubit certified bounds at large m when cost C is non-negligible.Source: §4 · p. 4
5 Heuristic scaling
Across low-rank and high-purity state scenarios, VQFE bounds beat SSFB using truncation sizes that grow linearly rather than exponentially with system size.
5 Heuristic scaling
- In both low-rank and high-purity scenarios, the minimum m needed for VQFE bounds to beat SSFB is O(n), with m* ≈2 for TFB and CCFB.The scaling study averages over 2000 random states ρ and σ for systems with n = 2, . . . , 7 qubits.Source: §5 · p. 4
- For high-purity ρ, m* also remains O(n), so the bounds can outperform SSFB without considering an exponentially scaling number of eigenvalues.This scenario uses states with exponentially decaying spectra leading to full rank but high purity.Source: §5 · p. 4
6 Example: Quantum phase transition
For an eight-spin cyclic Ising chain, the fidelity spectrum detects the transition near h = 1 and provides information beyond a single fidelity value. Its transition-location range tightens with m and beats SSFB at m = 3.
6 Example: Quantum phase transition
- The example uses thermal states of an N = 8 cyclic Ising chain at inverse temperature β = 10/J, comparing fields h and h + δh with δh = 0.01/J.The model is exactly solvable, and its thermal states can be exactly prepared on a quantum computer.Source: §6 · p. 4
- The truncated-fidelity bounds give a transition-field range that tightens monotonically with m and outperforms SSFB already at m = 3.Both SSFB and TFB show a pronounced dip near h = 1, indicating the zero-temperature transition.Source: §6 · p. 4
- Near h = 1, the m = 3 and 4 bounds show a discontinuity associated with a crossing between a uniform eigenstate and degenerate non-uniform symmetry-breaking states.The crossing changes the structure of the subspace spanned by the eigenvectors.Source: §6 · p. 4
- The fidelity spectrum also gives information about eigenvector closeness and state structure beyond SSFB or a single fidelity value.For example, the upper truncated-fidelity bounds are approximately 1 at m = 1 and 2.Source: §6 · p. 4
7 Complexity analysis
Low-Rank Fidelity Estimation is classically hard under standard complexity assumptions, even when one state is low rank. VQFE can solve it in polynomial time under specified rank, precision, and diagonalization-error conditions.
7 Complexity analysis
- VQFE does not require knowledge of the circuit descriptions that prepare ρ and σ.The problem definition nevertheless permits such descriptions, which is useful when considering classical algorithms.Source: §7 · p. 5
- Low-Rank Fidelity Estimation to inverse-polynomial precision is DQC1-hard, implying classical algorithms cannot efficiently solve it unless the polynomial hierarchy collapses to its second level.The problem allows ρ to be low rank or approximately low rank while σ is unrestricted.Source: §7 · p. 5
- For ε-low-rank ρ with polynomial m, VQFE runs in polynomial time for inverse-polynomial precision when mC and ζ are inverse-polynomial and ∥T∥ ≤ 1.This assumes access to a subroutine that approximately diagonalizes ρ.Source: §7 · p. 5
8 Conclusion
The paper concludes that its spectrum-truncation bounds and VQFE can support fidelity estimation, outperform established bounds in typical cases, and detect quantum phase transitions. It also identifies reconstruction effects and extensions to other quantities as open research directions.
8 Conclusion
- The authors report that their truncation-based bounds typically outperform sub- and super-fidelities and can detect quantum phase transitions.They propose VQFE to compute the bounds and anticipate its use in verifying and characterizing quantum-computer-prepared states.Source: §8 · p. 5
- The effect of maximum-likelihood reconstruction on the bounds remains to be analyzed because such procedures can bias estimated entanglement properties.This caveat concerns reconstruction of T when noise makes it non-positive.Source: §8 · p. 5
- The authors identify estimating distances between quantum operations and extending VQFE to sandwiched Rényi relative entropies as future directions.They note that the same eigenvalues of ρ and matrix elements of σ used by VQFE determine Dα(ρ||σ), with α = 1/2 corresponding to fidelity.Source: §8 · p. 5
A Proof of Proposition 1
- The proof of the truncated-fidelity bounds uses monotonicity under completely positive trace non-increasing maps for one side and strong concavity of fidelity for the other.The proof decomposes ρ into its truncated part and the remaining component.Source: A Proof of Proposition 1 · p. 9
B Proof of Proposition 2
- The proof establishes that truncated fidelity increases monotonically with m, while truncated generalized fidelity decreases monotonically with m.It derives these properties by expanding successive truncations and applying strong concavity and monotonicity under completely positive trace non-increasing maps.Source: B Proof of Proposition 2 · p. 9
C Proof of Proposition 3
These appendix proofs support certified cost-function bounds using ε-rank arguments and establish a certifiable triangular-inequality bound for arbitrary quantum states.
C Proof of Proposition 3
- Lemma 4 generalizes the main result of [52] beyond its ε = 0 special case, and the proof’s lemmas use the paper’s ε-rank notion.The ε-rank is defined by the smallest truncation rank whose discarded trace weight is at most ε.Source: C Proof of Proposition 3 · p. 9
- The proof bounds the ε-ranks of the positive and negative parts of ρ − σ by the ε-ranks of ρ and σ, respectively.It also notes that the bound on the fidelity difference holds for |∆F| by symmetry.Source: C Proof of Proposition 3 · pp. 10–11
- Proposition 3 gives certified cost-function bounds, with the proof deriving bounds through trace and Hilbert–Schmidt distance relations involving reduced rank.The appendix defines the reduced rank from the ranks of the two states and notes that the Hilbert-space dimension enters the distance relation.Source: C Proof of Proposition 3 · pp. 9, 11, 12
D Proof of Proposition 4
- Proposition 4 applies to arbitrary quantum states and derives certifiable triangular-inequality bounds from the distance measure’s triangle inequality and the truncated-fidelity bounds.The proof applies the triangle inequality through an intermediate state ρ′.Source: D Proof of Proposition 4 · pp. 12–13
E Proof of Proposition 5
The proof of Proposition 5 establishes DQC1-hardness by reducing a circuit-overlap approximation problem to Low-Rank Fidelity Estimation using Choi states.
E Proof of Proposition 5
- Low-Rank Fidelity Estimation to inverse-polynomial precision is DQC1-hard.The reduction encodes two circuits as Choi states and uses their fidelity to approximate the circuits’ Hilbert–Schmidt inner-product magnitude to inverse-polynomial precision.Source: E Proof of Proposition 5 · p. 13
- The reduction prepares the Choi states as pure states using the circuit and an entangling unitary.The two constructed states are prepared by applying the respective circuits to one half of a maximally entangled state.Source: E Proof of Proposition 5 · p. 13
F Proof of Proposition 6
Proposition 6 bounds VQFE’s runtime and fidelity-approximation error, with the error parameter tied to the ϵ-rank of ρ. The proof accounts for diagonalization and finite-sampling errors, and gives an alternative precision guarantee based on ρ′.
Alternative guarantee
- An alternative guarantee uses the experimentally measurable ϵ-rank of ρ′, so precision can be guaranteed without prior knowledge of ρ’s rank.This version does not require the promise that ρ is low rank.Source: F Proof of Proposition 6 · p. 14
Error analysis
- The proof combines bounds on the truncated-fidelity error with a probabilistic bound on finite-sampling error to bound the difference from the desired fidelity.The algorithm outputs an additive ±γ′-approximation with probability at least 1 − δ under the Lemma 6 precision definition.Source: F Proof of Proposition 6 · pp. 15–16
- The proof models diagonalization error as C = D_HS(ρ, ρ′), measures σ’s matrix elements in the approximate eigenbasis, and accounts for finite-sampling error in those estimates.Each needed entry is estimated using O(m4/ζ2 · log(2m2/δ)) Swap Tests; the simultaneous entrywise relative error is at most ζ/m2 with probability greater than 1 − δ.Source: F Proof of Proposition 6 · p. 15
Proposition 6
- With m = rankϵ(ρ), VQFE runs in eO(m6/ζ2) and outputs an additive ±γ-approximation of F(ρ, σ) with probability at least 1 − δ.The precision γ is defined by Eq. (83).Source: F Proof of Proposition 6 · p. 16