New directions in dynamical expectation estimation

• Existing approximation criteria optimize the state or observable in isolation, without accounting for the object used to compute the expectation value.Source: §1 · p. 1
• The proposed principle guides state and observable approximations by how their errors combine in the final expectation value.Source: §1 · p. 1
• The example forward loss combines expectation-value error with conventional state-only approximation error, with their relative influence controlled by γ and µ.The observable is propagated backward and held fixed during the update; this is an example objective, not a uniquely preferred one.Source: §1 · pp. 1–2
• Several orders of magnitude smaller errors than MPS-based circuit simulations and Pauli propagations are reported in the tested settings, while the sweep-like approach is not claimed to be optimal.The error relations provide an analytical rationale for aligning state and observable approximations to prioritize features relevant to the expectation value.Source: §1 · p. 2

• MPS-based simulations interleave evolution with bond-dimension-limited truncations, using truncated SVD or variational compression.Source: §2 · p. 2
• For normalized pure states, the state-compression objective can equivalently be expressed as maximizing fidelity.Source: §2 · p. 2
• Pauli propagation evolves the final observable backward; observable truncation can use MPO Schmidt truncation or discard high-Pauli-weight strings, while a hybrid method truncates forward states and backward observables independently.The observable is represented as a tensor network whose virtual bonds encode Pauli-string coefficients.Source: §2 · p. 2
• Across these methods, truncation criteria remain one-sided and do not account for how state and observable approximation errors combine in the final expectation value.Source: §2 · p. 2

3 Algorithm

• The formulation targets expectation values from quantum circuits and also applies when dynamics decompose into a sequence of local evolution maps.Source: §3 · p. 3

Algorithm 1

• Forward state and backward observable approximations are updated in opposing time sweeps, with maximum virtual bond dimensions χρ and χO.The algorithm describes forward and backward time sweeps; its backward loss reverses the roles of state and observable.Source: §3 · p. 3, Algorithm 1 · p. 3
• The backward loss combines expectation-value contraction error with Hilbert–Schmidt distance from the uncompressed observable target.The loss is ℓB(O; O⋆, ρ̄) = γ|Tr[ρ̄(O − O⋆)]|² + µ∥O − O⋆∥².Source: Algorithm 1 · p. 3
• Simple SVD-based sweeps are one possible way to construct the ansätze used by the algorithm.Source: Algorithm 1 · p. 3
• Setting γ = 0 recovers conventional state compression, while γ > 0 also drives the approximate expectation toward its uncompressed counterpart; the backward loss reverses state and observable roles.Loss-minimizing updates generally do not preserve unitarity, despite unitary underlying gates.Source: Algorithm 1 · p. 4

Rationale

• The γ-terms penalize local changes in the expectation value, while the µ-terms regularize the optimization toward approximations closer to the uncompressed target.The µ-term also helps limit residual norms entering the bound on R(i)₂.Source: §4 · pp. 4–5

Error relations

• Each contribution to R(i)₂ is bounded by a product of two approximation errors, whereas R(i)₁ contains only one approximation error per term.This distinction supplies the basis for the different error scaling of the two remainders.Source: §4 · p. 5

Numerical illustration

• Across 20 20-qubit barren plateau instances, Algorithm 1 suppresses |R(1)₁| relative to |R(0)₁|, while both |R(0)₂| and |R(1)₂| generally remain smaller than |R(0)₁|.The instances use depth 50 and maximum bond dimensions χρ = χO = 30.Source: §4 · p. 5, Figure 1 · p. 5

5.1 Implementation details

• The forward update minimizes ℓF, combining conventional state compression through µ with expectation-value error at each time slice through γ and the backward-propagated observable.The backward sweep reverses direction and represents the adjoint action of Ut in the Pauli basis.Source: §5.1 · p. 6, Figure 2 · p. 6
• For a fixed number of local update iterations, a full sweep has time and memory costs scaling as poly(χρ, χO, n, L).The benchmarks use barren plateau circuits with single-qubit Pauli rotations and a nearest-neighbor controlled-Z ladder.Source: §5.1 · p. 7, §5.2 · p. 7

5.2 Benchmarks

• At equal bond dimensions, the proposed algorithm yields expectation-value errors two to three orders of magnitude smaller than variational state compression.The comparison uses 500 randomly generated 30-qubit barren plateau circuits.Source: §5.2 · p. 7

6 Conclusion

• The time-sweep algorithm jointly guides state and observable approximations by how their errors combine in the final expectation value.The considered loss functions are heuristic choices whose optimal form remains open.Source: §6 · p. 7
• Figure 3 reports relative errors averaged over 500 barren plateau circuit instances with 30 qubits, with Algorithm 1's backward-observable bond dimension matched to the forward MPS.The caption identifies the comparison scope but does not itself establish which method performs best.Source: Figure 3 · p. 8

A Detailed derivation of the error decomposition

• The decomposition applies to approximations at both stages: i=0 denotes initial SVD-based approximations, and i=1 denotes those after the forward and backward sweeps.Source: A Detailed derivation of the error… · p. 10
• The derivation telescopes successive differences while retaining their signs, and does not assume monotonicity of q(i).Source: A Detailed derivation of the error… · p. 10
• Accumulated state and observable errors are expressed as sums of propagated local defects, using unitary propagation from one time slice to another.The backward propagation defect differs from accumulated observable error; the derivation also defines a forward propagation defect.Source: A Detailed derivation of the error… · pp. 10–12
• The expectation-value error decomposes into R(i)_1 + R(i)_2, with the two contributions equal as sums even though their individual terms need not coincide.The derivation uses propagation relations and endpoint conditions, not exact solutions to local compression problems.Source: A Detailed derivation of the error… · p. 12