Anastasiou et al. describe the change as lifting exactly one rule: this is "a modified version of the ADAPT-VQE algorithm in which the one-operator-at-a-time rule is lifted to allow for the addition of multiple operators with disjoint supports in each iteration". What it buys is stated without a number and with its limits named — the result is "denser but significantly shallower circuits, without increasing the number of CNOT gates or variational parameters", and "its advantage over the original algorithm in terms of circuit depths increases with the system size". It also cuts the measurement overhead ADAPT pays between rounds: "the expensive step of measuring the energy gradient with respect to each candidate unitary at each iteration is performed only a fraction of the time compared to ADAPT-VQE", because fewer rounds are needed once a round may add more than one operator. The motivation is hardware rather than accuracy — adaptive algorithms "are not yet viable due, in large part, to the severe coherence time limitations on current devices".

The Hamiltonian whose ground state is wanted, together with whatever structure is to be respected — particle number, spin, point-group symmetry, a reference determinant — and the connectivity and native gate set of the device the family has to run on.

1. Hamiltonian whose eigenvalues are wanted → Parameterised circuit familysort by gradient, pack disjoint-support ops

   Each iteration starts from ADAPT's per-operator gradient, $\partial E/\partial\theta_i|_{\theta_i=0} = \langle\Psi^{(k)}|[H,P_i]|\Psi^{(k)}\rangle$, valid under assumption: antihermiticity $P_i^\dagger=-P_i$ of every pool generator. TETRIS keeps this one gradient measurement per iteration but replaces the single append with subroutine 4': sort all pool operators by gradient norm, descending, then repeatedly append the next-largest-gradient operator whose qubit support is disjoint from every operator already added this iteration, each new parameter set to zero, approximation: continuing by this greedy descending-gradient order rather than jointly choosing the disjoint subset that lowers the energy most, until every qubit is covered or no disjoint-support operator with nonzero gradient remains.

   approximationassumption

A circuit family with a fixed structure and free real parameters, together with the number of those parameters — which is the size of the classical search problem handed to the next layer.

• Estimate an observable

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

  What TETRIS hands down is a single state — the current ansatz $|\Psi^{(k)}\rangle$ at its optimized parameters — together with the whole operator pool; what comes back is one number per pool operator, and the quantity the selection actually uses is its norm, $|g_i^{(k)}| = |\langle\Psi^{(k)}|[H,P_i]|\Psi^{(k)}\rangle|$. ADAPT-VQE consumes only $\arg\max_i |g_i^{(k)}|$ from that returned vector; TETRIS's subroutine 4' consumes the ranking itself, descending past the maximum and keeping each entry whose qubit support is disjoint from those already kept, so estimates well below the largest decide what enters the layer. Reading deeper costs the layer no extra calls: the gradients "need only be measured once per iteration of the algorithm as usual". approximation: every operator packed into one iteration is ranked by a gradient estimated on $|\Psi^{(k)}\rangle$, the state before any of that iteration's operators were appended — none is re-estimated as the layer fills

  approximation

given  Hamiltonian H on Q qubits (already qubit-mapped, e.g. Jordan-Wigner, Sec. II);
           operator pool P = {P_i}, antihermitian generators -- typically the qubit
           or QE pool (Eqs. 11-13), chosen over the fermionic one (Eqs. 7-8) because
           their operators touch fewer qubits each;  reference state |Psi_ref>,
           usually Hartree-Fock;  gradient-norm convergence threshold eps

requires  P_i^dagger = -P_i  for every pool operator P_i
    # antihermiticity is what turns the raw energy derivative into the
    #   commutator expectation value used below (Sec. II B, at Eq. 6)

# --- outer ADAPT loop: grow the ansatz one layer at a time (Sec. II B, steps 1-6) ---
Psi^(0) = |Psi_ref>;  ansatz = [ ]  (ordered list of (P_i, theta_i) pairs);  k = 0
loop
    # --- one gradient sweep, the whole pool handed to observable-estimation ---
    {g_i^(k)}_{P_i in P} = observable-estimation(H, Psi^(k), P)
        # g_i^(k) = <Psi^(k)| [H, P_i] |Psi^(k)>                          (Eq. 6)
        # measured once per iteration, however many operators this iteration
        #   goes on to pack below: 'this can be done without measuring any
        #   additional gradients compared to ordinary ADAPT-VQE since the
        #   gradient of each pool operator need only be measured once per
        #   iteration of the algorithm as usual'

    if sqrt(sum_i (g_i^(k))^2) < eps:                                    # step 3
        return ansatz
            # TETRIS's own text says only 'the pool gradient norm' -- this
            #   step is inherited unchanged from ADAPT-VQE (ref [15]), which
            #   resolves it explicitly as the L2 norm of the gradient
            #   VECTOR across the whole pool, not any single |g_i^(k)|
            # 'ADAPT-VQE has converged, and the algorithm terminates'
            #   (Sec. II B, step 3) -- TETRIS reuses this criterion as-is
            # the paper's own numerics used eps = 1e-7 (Sec. III, opening
            #   paragraph) -- a run choice, not part of the algorithm itself

    # --- subroutine 4': pack disjoint-support operators into one layer ---------
    order = pool operators sorted by |g_i^(k)|, descending            # step 4'(a)
    covered = {}                   # union of qubit supports used this iteration
    layer = [ ]
    loop
        pick the first P_i in order with support(P_i) disjoint from covered
                                 and g_i^(k) != 0                       # step 4'(b)
        if no such P_i exists:  break                                  # step 4'(d)
        append (P_i, theta_i = 0) to layer
        covered = covered union support(P_i)
        if covered = {all Q qubits}:  break                            # step 4'(c)
    # the order above is fixed by gradient magnitude alone -- greedy, NOT a
    #   joint search for the disjoint subset that would lower the energy most

    ansatz = ansatz ++ layer
        # every pair in layer sits on qubits none of the others touch, so
        #   the whole layer can be 'implemented simultaneously in the
        #   circuit' (Sec. IV, point a) -- one packed layer per iteration,
        #   in place of ADAPT-VQE's single operator

    (ansatz, Psi^(k)) = optimize(ansatz)
        # 'perform a VQE subroutine to update all parameters in the
        #   current ansatz' (Sec. II B, step 5) -- the optimum overwrites
        #   each theta_i stored in ansatz itself, not just Psi^(k), since
        #   the return line below reads theta_i off of ansatz directly
        # step 5 does not specify how the parameters are initialised;
        #   Sec. III D tests recycling the previous round's optimum for
        #   pre-existing operators ('warm start') against cold-start and
        #   300 random draws and reports it as the usual, effective choice
        #   -- an empirical finding, not part of the step-5 definition
    k = k + 1

return  ansatz, read as  prod_layers ( prod_{(P_i,theta_i) in layer} exp(theta_i P_i) )
        applied to |Psi_ref>  --  a parameterised circuit family, each layer's
        gates scheduled simultaneously, handed to the calling VQE loop

Anastasiou, Chen, Mayhall, Barnes and Economou price TETRIS-ADAPT-VQE only relative to ADAPT-VQE. Measured at ADAPT-VQE convergence, averaged over an unenumerated set of bond lengths with failed geometries excluded and never counted: the circuit-depth ratio runs $1.58$ to $2.73$ across H4, LiH, H6 and BeH2 on 8 to 14 qubits, reported separately for the qubit and QE pools and growing with system size (Table I), and pool-gradient measurements fall 53% to 70%, pools not resolved and no operator grouping for simultaneous measurement assumed (Table II). CNOT and variational-parameter counts are, in the authors' words, "roughly the same" as ADAPT-VQE's, not smaller. Depths assume all-to-all connectivity, Qiskit optimization level 1, and ADAPT-VQE given the same concurrency. Each iteration still measures $\langle \Psi|[H,P_i]|\Psi \rangle$ for every pool operator; packing adds no gradients. Nothing bounds the iteration count — the authors merely expect the round reduction to be "roughly $\lfloor Q/4 \rfloor$" for $Q$ qubits, assuming a Jordan-Wigner, double-excitation-dominated ansatz — and the running text's only iteration count is 13, for H4 at 3.0 Angstrom. Absolute depths and CNOT counts appear only in Figs. 4 and 5.

• TETRIS-ADAPT-VQE · Qiskit

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  Operators with disjoint support are packed into the same adaptive layer to reduce circuit depth.

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• TETRIS-ADAPT-VQE

  Operators with disjoint support are packed into the same adaptive layer to reduce circuit depth.