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MethodLayer 0

Quantum subspace expansion

Stop optimising and start measuring: take the state the ground-state run already produced, measure matrix elements in a small space of operators applied to it, and let a classical solver diagonalise the little matrix that results. The extra accuracy is bought with classical work and more measurements, not with more coherence.

Takes

A Hermitian H reachable as a sum of terms that can be measured separately; a statement of which state is wanted — an index k, a symmetry sector, or a target energy to sit nearest; for most of the methods here, the ground state already solved, and for the deflation route every lower state as well; a target additive error and a shot budget.

Returns

A scalar estimate — of that eigenvalue, or of the gap between it and the ground state — together with the run budget it consumed. Some methods return a whole low-energy subspace at once and others return one state per run, and that difference is a cost, not a convenience.

Same contract as the slot it fills.

This one, drawn

Quantum subspace expansionVariational quantum deflation · 3 parts insideVariational quantum deflation · 3 parts inside — click the line to open it hereSubspace-search variational eigensolver · 3 parts insideSubspace-search variational eigensolver · 3 parts inside — click the line to open it hereQuantum subspace expansionQuantum equation of motionFolded-spectrum variational eigensolver · 3 parts insideFolded-spectrum variational eigensolver · 3 parts inside — click the line to open it herePenalty-constrained variational eigensolver · 3 parts inside, openPenalty-constrained variational eigensolver · 3 parts inside, openPenalty-constrained variational eigensolver · 3 parts inside, open — click the line to close itChoose a parameterised trial state · 13 ways throughChoose a parameterised trial state · 13 ways through — click the line to open it hereMinimise the objective over the parameters · 6 ways throughMinimise the objective over the parameters · 6 ways through — click the line to open it hereEstimate an observable · 4 ways throughEstimate an observable · 4 ways through — click the line to open it hereMultistate contracted variational eigensolver · 3 parts insideMultistate contracted variational eigensolver · 3 parts inside — click the line to open it hereVariational quantum deflation · 3 parts inside — click the name to read about itVQDSubspace-search variational eigensolver · 3 parts inside — click the name to read about itSSVQEQuantum subspace expansion — click the name to read about itSubspace expansionQuantum equation of motion — click the name to read about itqEOMFolded-spectrum variational eigensolver · 3 parts inside — click the name to read about itFolded spectrumPenalty-constrained variational eigensolver · 3 parts inside, open — click the name to close itPenalty termsChoose a parameterised trial state · 13 ways through — click the name to read about itChoose an ansatzMinimise the objective over the parameters · 6 ways through — click the name to read about itMinimise the objectiveEstimate an observable · 4 ways through — click the name to read about itEstimate an observableMultistate contracted variational eigensolver · 3 parts inside — click the name to read about itMC-VQEHamiltonian whose excited state is wanted — you start hereNumber with an error bar — you finish hereParameterised circuit family — what one part hands to the nextState you can prepare — what one part hands to the next

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From Hamiltonian whose excited state is wanted to Number with an error bar

A circle is an object you are holding. This method is drawn heavier, opened into its own steps; the other lines between the same two ends are the alternatives recorded for the same slot. Circles are named on hover, and each one is a link.

7 lines here have ways through that this figure does not open. The map opens them in place. See it on the map

What it fills

  • Estimate an excited-state energy

    Given a Hamiltonian you can query and a statement of which state above the lowest one is wanted, return an estimate of that state's energy — or of its distance from the ground state, which is the quantity an experiment usually measures. Whether the answer comes back as an absolute energy or as an excitation energy is a property of the method, and the two are not interchangeable.

When it applies

McClean et al. state the shape of the guarantee rather than a bound: variational approaches "fit in a more general hierarchy of measurement and classical computation that allows one to obtain increasingly accurate solutions with additional classical resources", and they "demonstrate numerically on a sample electronic system that this method both allows for the accurate determination of excited electronic states as well as reduces the impact of decoherence, without using any additional quantum coherence time or formal error correction codes". The decoherence claim is separate and is argued from an "exactly solvable channel model of variational state preparation". What the abstract does NOT give is the mechanism at the level a reader would need to implement it — which operators span the expansion, and which matrix elements have to be measured — and neither is invented here; both sit in the body of the paper, and the honest summary of the abstract is the trade it names, not a construction it does not state.

Requires

These do not move the route along. The method needs each of them alongside its own work, and the cost of getting them is part of what the method costs.

  • Estimate a Hamiltonian's ground-state energy

    Given a Hamiltonian you can query and some way of preparing trial states, return an estimate of its lowest eigenvalue to a stated additive error. The state achieving that energy may or may not come back with the number; almost every application wants the number.

    The route enters this slot with an ordinary VQE loop. A parameterised ∣Ψ(θ⃗)⟩|\Psi(\vec{\theta})\rangle is prepared and its parameters driven to the Rayleigh-Ritz minimum min⁡θ⃗⟨H⟩(θ⃗)=⟨Ψ(θ⃗)∣H∣Ψ(θ⃗)⟩/⟨Ψ(θ⃗)∣Ψ(θ⃗)⟩\min_{\vec{\theta}} \langle H \rangle(\vec{\theta}) = \langle \Psi(\vec{\theta})|H|\Psi(\vec{\theta})\rangle / \langle \Psi(\vec{\theta})|\Psi(\vec{\theta})\rangle, the energy read out by repeated state preparation and partial tomography as the contraction ⟨H⟩=∑ikhik 1Dki+∑ijklhijkl 2Dlkij\langle H \rangle = \sum_{ik} h_{ik}\,{}^1D^i_k + \sum_{ijkl} h_{ijkl}\,{}^2D^{ij}_{lk}. McClean et al. model the preparation as a channel rather than a unitary: minimising Tr[(∑iKi∣Ψ⟩⟨Ψ∣Ki†)H]\mathrm{Tr}[(\sum_i K_i |\Psi\rangle\langle\Psi| K_i^\dagger) H] is equivalent to solving H′∣Ψ⟩=E∣Ψ⟩H'|\Psi\rangle = E|\Psi\rangle on the transformed H′=∑iKi†HKiH' = \sum_i K_i^\dagger H K_i for the lowest eigenvalue and eigenvector pair. assumption: the model prepares an arbitrary pure state, so it fixes the optimal possible performance separately from ansatz choice or experimental protocol Symmetry can be imposed by a penalty H→H+∑iλi(Oi−oi)2H \to H + \sum_i \lambda_i (O_i - o_i)^2 approximation: exact only as λi→∞\lambda_i \to \infty, and only if the parameterisation can satisfy the symmetry at all; the authors call it prohibitively expensive and warn it may not generalise to excited states of a different symmetry. More than the energy leaves this slot: the 1- and 2-RDM already measured are the reference the next hop expands about, which then buys the 3- and 4-RDM with a polynomial number of additional measurements.

    approximationassumption

  • Estimate an observable

    Given the ability to prepare ∣ψ⟩|ψ⟩ and a description of an observable OO, return a classical scalar within ε\varepsilon of ⟨O⟩⟨O⟩ at confidence 1−δ1−δ. The state is never returned; only the number is.

    Here the route stops optimising. It expands about ρ\rho, the state at the exit of the channel, in the vectors ai†aj∣Ψ⟩a_i^\dagger a_j |\Psi\rangle. Partial tomography supplies the elements within that expansion, Oklij=Tr[(ai†aj)†O ak†al ρ]O^{ij}_{kl} = \mathrm{Tr}[(a_i^\dagger a_j)^\dagger O\, a_k^\dagger a_l\, \rho] with metric Sklij=δik 1Dlj−2 2DlijkS^{ij}_{kl} = \delta_{ik}\,{}^1D^j_l - 2\,{}^2D^{jk}_{li}, and those coupling it to the reference, Ogij=Tr[(ai†aj)†Oρ]O^{ij}_g = \mathrm{Tr}[(a_i^\dagger a_j)^\dagger O \rho] with Sgij=1DijS^{ij}_g = {}^1D^j_i; that gg row and column border the matrices, so the reference sits inside the diagonalised space. A one-body operator reaches at most the 3-RDM, a two-body V=∑pqrsap†aq†arasV = \sum_{pqrs} a_p^\dagger a_q^\dagger a_r a_s the 4-RDM through a −24 4Dsrlijkpq-24\,{}^4D^{jkpq}_{srli} term. The classical step solves HLRC=SLRCEH_{\mathrm{LR}} C = S_{\mathrm{LR}} C E for the ground and excited states; symmetry is imposed here with no λi→∞\lambda_i \to \infty limit, by building OLRO_{\mathrm{LR}} in the same subspace and projecting onto its eigenvectors of the wanted eigenvalue. approximation: truncating the hierarchy Bfk\mathcal{B}^k_f at k=1k=1 spans only the linear-response space, justified as the dominant contribution in a linear response theory of local time-dependent perturbations approximation: the commutator form Hklij=⟨Ψ∣(ai†aj)†[H,ak†al]∣Ψ⟩+Eg⟨Ψ∣(ai†aj)†ak†al∣Ψ⟩H^{ij}_{kl} = \langle \Psi|(a_i^\dagger a_j)^\dagger [H, a_k^\dagger a_l]|\Psi\rangle + E_g \langle \Psi|(a_i^\dagger a_j)^\dagger a_k^\dagger a_l|\Psi\rangle removes the 4-RDM at O(M6)O(M^6) measured terms and is exact only when ∣Ψ⟩|\Psi\rangle is exact; ZC further sets the 3-particle cumulant 3Δ=0{}^3\Delta = 0 approximation: ZA instead starts from the un-reduced 4-RDM and reconstructs it from the 2-RDM alone, setting 4Δ=3Δ=0{}^4\Delta = {}^3\Delta = 0

    approximation

Example

given  a normalized reference state |Psi> already produced by a prior VQE run
       that minimized <H>(theta) over the ansatz parameters (Eq. 18) -- this
       method does not run that optimization, it consumes its output
       the reference's 1- and 2-RDM, 1D^i_k and 2D^ij_kl (Eq. 3-4), already
       measured as part of that VQE run
       M spin-orbitals (JW-encoded qubits), so excitation indices i,j range
       over [1, M]
       optionally, an operator O (a symmetry operator, or any observable
       whose value is wanted in the excited states) expressed as a sum of
       one- and two-body fermion operators, as H itself is (Eq. 10)

requires  |Psi> is the single state the subspace is built ABOUT, not a family
    of trial states -- the paper demonstrates the k=1 truncation's accuracy
    from an exact ground-state reference (Fig. 4) and separately from an
    approximate, channel-degraded one (Fig. 5, amplitude/phase damping); it
    states no general bound on truncation error for an arbitrary reference

# --- build the k=1 excitation subspace -------------------------------------
Bf1 = { a_i^dagger a_j |Psi>  :  i, j in [1, M] }                (Eq. 6, Eq. 8 at k=1)
    # justified only as the dominant contribution in a linear-response theory
    #   of local time-dependent perturbations to the system -- that result is
    #   cited to a separate reference, not derived in this paper       (ref [28])
    # Bf1 is level k=1 of a hierarchy Bf^k; k=Ne would span the whole
    #   Ne-fermion space and make the classical step exact diagonalization,
    #   but its cost then scales exponentially in system size      (text after Eq. 8)

# --- measure the subspace's Hamiltonian and overlap by partial tomography --
for each (i,j), (k,l) in [1,M] x [1,M]:
    H^ij_kl = <Psi| (a_i^dagger a_j)^dagger H a_k^dagger a_l |Psi>          (Eq. 45, O = H)
    S^ij_kl = delta_ik * 1D^j_l  -  2 * 2D^jk_li                            (Eq. 47)
    # S is the overlap/metric of the (non-orthogonal) excitation vectors --
    #   it is NOT the identity, which is exactly why the next step is a
    #   GENERALIZED eigenvalue problem, not a plain diagonalization
    # S alone is fixed by the 1- and 2-RDM (Eq. 47); H is NOT -- a one-body
    #   term of H reaches the 3-RDM, a two-body term the 4-RDM (matrix
    #   elements given explicitly, Eq. 48-51), and determining these costs a
    #   polynomial number of measurements beyond what the ground-state VQE
    #   run already took                                        (text after Eq. 5)

# --- solve the subspace problem classically ---------------------------------
solve  H_LR C = S_LR C E   for eigenpairs (C, E)                    (Eq. 7 / Eq. 21)
    # E's lowest eigenvalue approximates the ground state within Bf1; its
    #   higher eigenvalues are the excited-state estimates this method is for

# --- optional: enforce an exact symmetry in the subspace --------------------
if a symmetry operator O with target eigenvalue o is supplied:
    build  O^ij_kl  the same way as H^ij_kl above
    project onto the eigenvectors of O_LR carrying eigenvalue o
    diagonalize H_LR restricted to that projected subspace          (Supp. Sec. C)
    # exact for any finite subspace here -- unlike the penalty
    #   H -> H + sum_i lambda_i (O_i - o_i)^2 used to impose a symmetry on
    #   the reference state itself, which the paper notes is only exact as
    #   lambda_i -> infinity and "may not generalize well to excited states
    #   if they are of a different symmetry than the ground state"   (main text, Eq. 20)
    # the paper gives this projection only in prose, not as an explicit
    #   projector formula                                            (Supp. Sec. C)

# --- optional variants that avoid measuring the 4-RDM -----------------------
# the fermionic QSE above is exact within Bf1 but, in the paper's own words,
#   "can still be prohibitively expensive in the number of measurements
#   required"; both variants below are approximations aimed at that   (Supp. Sec. I)
# ZC: H^ij_kl ~= <Psi|(a_i^dagger a_j)^dagger [H, a_k^dagger a_l]|Psi>
#                + Eg <Psi|(a_i^dagger a_j)^dagger a_k^dagger a_l|Psi>        (Eq. 59)
#     exact only when |Psi> is the exact ground state (Eg its eigenvalue);
#     the commutator drops the expression to at most the 3-RDM, O(M^6)
#     terms; setting the irreducible 3-particle cumulant 3Delta = 0 removes
#     even that, needing nothing beyond the ORIGINAL ground-state 2-RDM
# ZA: skips the commutator and reconstructs the 4-RDM from the 2-RDM alone,
#     assuming 4Delta = 3Delta = 0
# the two differ in ACCURACY, not in measurement cost: the paper states
#   "both methods require only the original measurements used for the ground
#   state", with ZC at "extremely high accuracy" while ZA is "qualitatively
#   correct in some cases but produces sub-variational solutions in others";
#   the extra structure of ZC is what buys the accuracy         (Fig. 7 caption)
# both accuracy comparisons are a single numerical demonstration on H2, not
#   a bound                                                            (Fig. 7)

return  the eigenpairs (C, E) -- E holds ground- and excited-state energy
    estimates within Bf1, C the coefficients of each state in the
    a_i^dagger a_j |Psi> basis
# the identical C, with O^ij_kl computed in place of H^ij_kl, reads off <O>
#   for any of these states with no new optimization -- the mechanism this
#   method is reused for whenever the wanted quantity is not the energy

Cost, as the source states it

McClean, Schwartz, Carter and de Jong buy excited states with measurements, not coherence. At the linear-response level k=1k=1 of the hierarchy of Eq. (8), whose kk index pairs each range over [1,M][1,M] spin-orbitals — the paper writes the index ranges, not the counts — that is M2M^2 vectors ai†aj∣Ψ⟩a_i^\dagger a_j |\Psi\rangle, M4M^4 labelled matrix elements OklijO^{ij}_{kl}, and one generalized eigenvalue problem HLRC=SLRCEH_{LR}C = S_{LR}CE (Eq. 7) solved classically. The 1- and 2-RDM come from the ground-state run; for the extra 3- and 4-RDM the authors claim only "a polynomial number of additional measurements", never an exponent. Their one explicit count is the commutator form of Eq. (59), which "depends at most on the 3-RDM requiring only O(M6)O(M^6) terms to be measured", and is exact only when ∣Ψ⟩|\Psi\rangle is exact; the ZC and ZA cumulant approximations need nothing beyond the ground-state 2-RDM. No shot count or precision bound is given. Truncated levels the authors call "efficient"; at k=Nek=N_e the classical diagonalization "scales exponentially in the size of the system".

Implementations

Nobody has written one up yet. That is a gap in this record, not a statement that the method has never been run — the paper register already records, per paper, which sources report numerics or a hardware run.

What it needs

Every step this method names is listed under Requires above. It walks its own span in one hop and calls out to the rest — that is a fact about the recorded route, not a claim that the span is simple.

Other ways to fill the same slot

Different approaches

  • Variational quantum deflation

    Find the ground state first, then run the same variational search again with a term that punishes overlap with every state already found. Each state is reached by pushing the search off the ones below it, so they have to be found in order and each one costs another pass through the whole loop.

  • Subspace-search variational eigensolver

    Send several mutually orthogonal input states through one parameterised circuit and minimise their energies together. A unitary keeps orthogonal inputs orthogonal, so the whole low-energy subspace comes out of a single optimisation — no earlier state to deflate, and no ancilla to test overlaps with.

  • Quantum equation of motion

    Ask for the gaps directly rather than for two energies to subtract. Measure the matrix elements the classical equation-of-motion formalism needs between excitation operators on the ground state, and solve its generalised eigenvalue problem classically; what comes back is an excitation energy, which is the quantity a spectrum is made of.

  • Folded-spectrum variational eigensolver

    Point the same search somewhere other than the bottom. Minimising the variance around a chosen energy makes every eigenstate a minimum and the one nearest that energy the reachable one — so a state can be found without knowing its index, and the bill arrives as a squared Hamiltonian with far more terms to measure.

  • Penalty-constrained variational eigensolver

    Add a term to the objective that punishes the trial state for leaving the symmetry sector you asked for, and the ordinary ground-state search returns that sector's lowest state — an excited state of the whole Hamiltonian whenever the sector is not the one the ground state lives in. Which penalty is used matters: one common form is proved not to work.

  • Multistate contracted variational eigensolver

    Optimise one circuit for several states at once and read off the transitions between them — both the energy of each transition and the oscillator strength that says how strongly light drives it. The answer is a spectrum with intensities, which is what an absorption experiment actually produces.

In the Atlas

  • Quantum subspace expansion

    Measured response operators around a VQE state define a generalized eigenproblem for excitations and mitigation.

Sources