Efficient initial states for quantum phase estimation: overlap that classical heuristics cannot match
Difficulty 59.5: a complexity-flavoured question about physical Hamiltonians with no route to a proof and evidence that so far leans one way. Application value 42: it decides where quantum-chemistry effort is worthwhile. Leverage 54: resolves the central open question in the chemistry advantage debate.
The problem
A quantum computer needs a good first guess of a molecule's state; if one exists, would ordinary computers manage too?
- 1Phase estimation needs a good start
- 2Overlap shrinks with system size
- 3Better guesses exist classically
- 4If guess is easy, is energy easy?
- 5Open: separation or collapse
Setup. Quantum phase estimation applied to an input state returns the eigenvalue with probability . To get the ground-state energy with constant probability one needs , preparable in polynomial time. In words: the starting guess must contain a not-too-small share of the true ground state.
Built from
earlier problems → this oneThe problems this one is built from: progress or tools it needs (builds on), special cases that come first (generalizes), and weaker results on the way (implies). On the right, what builds on it.
- Start with
- Efficient initial states for quantum phase estimation: overlap that classical heuristics cannot matchThis problem
- Then
- A verified quantum advantage for a chemically relevant ground-state energyComputational Chemistry · D65.5
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Coupled-cluster theory and the CCSD(T) gold standardAlgorithm
- Quantum algorithmsAlgorithm
- Step 3
- Step 4
- Quantum computing for chemistryFrontier · host
- Then
- Efficient initial states for quantum phase estimation: overlap that classical heuristics cannot matchThe problem
State of the art
- Orthogonality catastrophe. A single Hartree–Fock determinant has overlap with the exact ground state that decays exponentially with system size for extended or strongly correlated systems. Tubman et al. (arXiv:1809.05523, 2018) show that sparse multideterminant states (selected CI) postpone this decay, and Fomichev et al. (PRX Quantum 2024; arXiv:2310.18410) develop and benchmark state-preparation routines including matrix-product-state and sparse-determinant loading. Berry, Tong et al. (arXiv:2409.11748, 2024) give a rapid procedure for strongly correlated molecules with improved gate cost.
- Assessment. Lee et al. (Nat. Commun. 2023) found, for the systems studied, that when efficient state preparation was available the classical heuristics also succeeded, and concluded that generic exponential advantage for ground-state chemistry is not supported by present evidence (polynomial speedups remain possible). They do not claim a proof.
- Complexity. Gharibian and Le Gall (arXiv:2111.09079; STOC 2022) study dequantization of the quantum singular value transformation and the complexity of ground-state problems with guiding states, with applications to quantum chemistry and the quantum PCP conjecture.
- FeMoco overlap. In the Zhai et al. FeMoco preprint (arXiv:2601.04621), the overlap between a filtered broken-symmetry determinant and a bond-dimension-18,000 DMRG state is reported as about 0.45 (their interpretation: overlap with the low-energy manifold); this is a preprint claim.
- Open. No family with a proven or convincingly demonstrated efficient-quantum / hard-classical separation exists for molecular Hamiltonians.
Realistic applications
Determines whether fault-tolerant phase estimation offers exponential or merely polynomial advantage for molecules, which sets the value of chemistry as a quantum application and guides where algorithm and hardware effort goes.
What counts as solved
A peer-reviewed result that either (a) gives a molecular family with provable or convincingly validated efficient state preparation and an accepted classical-hardness argument, or (b) gives a proof or strong theoretical argument that efficient state preparation implies classical tractability for physical molecular Hamiltonians, accepted by experts in both communities.
How it gets verified
Expert reviewOnly expert review of the argument is possible today.
The target is either a molecular family with provable or convincingly validated efficient state preparation and an accepted classical-hardness argument, or a proof that efficient state preparation implies classical tractability. This is a complexity-theoretic argument for physical Hamiltonians, so experts review it; numerical overlap studies on small systems support but do not settle it.
Claims under verification
+ add a claimNo claimed results on the Verification board for this problem yet.
Where the work is
Mostly theory, with numerical studies on molecular families beyond exact diagonalisation.
Bottlenecks
ranked by votes- #1TheoryRelating state-preparation efficiency to classical tractability
No theorem links the structure enabling efficient preparation to classical solvability for physical Hamiltonians.
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Connected problems
Progress
- Sep 18, 2024Rapid state preparationsource
Berry et al. give faster initial-state preparation for strongly correlated molecules (arXiv date).
- Apr 5, 2023Assessment of the evidencesource
Lee et al. conclude evidence for exponential advantage is lacking (date approximate).
- Nov 17, 2021Dequantization and guiding statessource
Gharibian and Le Gall relate dequantization to chemistry and quantum PCP (arXiv date).
- Sep 14, 2018Postponing the orthogonality catastrophesource
Tubman et al. show sparse multideterminant states keep overlap larger for longer (arXiv date).