TETRIS-ADAPT-VQE: Batched Adaptive Ansatz

• TETRIS-ADAPT-VQE is an adaptive variational quantum eigensolver that replaces the serial one-operator-at-a-time rule with a batched strategy appending multiple disjoint operators per iteration.
• It constructs denser yet significantly shallower circuits, reducing both adaptive iterations and full pool-gradient measurement rounds by up to 70% in benchmark simulations.
• The method is pool-independent and enhances expressivity for electronic structure and quantum dynamics simulations by leveraging disjoint operator supports for efficient ansatz growth.

TETRIS-ADAPT-VQE is an adaptive variational quantum eigensolver in which ADAPT-VQE’s one-operator-at-a-time growth rule is replaced by a batched rule that appends multiple high-gradient operators per adaptive iteration, provided their qubit supports are disjoint. Introduced as “Tiling Efficient Trial circuits with Rotations Implemented Simultaneously,” the method keeps ADAPT-VQE’s gradient-driven, problem-tailored ansatz construction, but aims to produce denser and significantly shallower circuits, reduce the number of adaptive iterations, and decrease the frequency of full pool-gradient measurement rounds (Anastasiou et al., 2022).

1. Position within the adaptive VQE family

TETRIS-ADAPT-VQE emerged from the same design pressures that motivated ADAPT-VQE more broadly: fixed ansätze such as low-order UCCSD are often either too rigid for strongly correlated regimes or too deep for NISQ execution, whereas adaptive constructions can tailor the operator content to the target Hamiltonian. Standard ADAPT-VQE builds the ansatz iteratively from a predefined operator pool, using energy-gradient information to decide which generator to append next. Benchmark studies had already established two relevant baseline facts: ADAPT-VQE is typically more robust than fixed-ansatz VQE with respect to optimizer choice, but it pays for this robustness with substantial operator-screening overhead because the full pool must be re-evaluated at every iteration (Claudino et al., 2020).

Within that landscape, TETRIS does not redefine the VQE objective and does not replace the ADAPT commutator-gradient logic. Its novelty is architectural. The method observes that ordinary ADAPT-VQE often appends an operator acting on only a few qubits while leaving many qubits idle in the same circuit layer. TETRIS therefore treats those unused qubits as an opportunity for concurrent ansatz growth. This preserves the adaptive character of ADAPT-VQE while changing the growth granularity from a serial list to a layer-building procedure (Anastasiou et al., 2022).

The motivation is also consistent with broader analyses of ADAPT behavior. ADAPT-VQE’s warm-started, gradient-informed growth tends to avoid the worst effects of rough parameter landscapes and barren-plateau-like regions along its actual optimization trajectory, but those advantages do not by themselves solve the depth and measurement costs caused by strictly serial ansatz growth (Grimsley et al., 2022). TETRIS should therefore be read as a structural modification of ADAPT’s growth rule rather than as a replacement of the underlying adaptive principle.

2. Formal algorithmic structure

The original TETRIS formulation is presented for molecular electronic structure with Hamiltonian

$$\hat{H}=\sum_{p,q} h_{pq}a_q^{\dagger}a_p+\frac{1}{2}\sum_{p,q,r,s} h_{pqrs}a_p^{\dagger}a_q^{\dagger}a_sa_r,$$

using Jordan–Wigner encoding in the reported simulations (Anastasiou et al., 2022).

As in ADAPT-VQE, the ansatz after adaptive growth has the generic product form

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$

and the optimization target remains the standard variational energy

$$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$

Operator selection is based on the usual ADAPT gradient signal. For a pool operator $P_i$ at adaptive iteration $k+1$, one evaluates

$$\frac{\partial E}{\partial \theta_i}\bigg|_{\theta_i=0} = \left[ \frac{\partial}{\partial \theta_i} \bra{\Psi^{(k)}}e^{-\theta_iP_i}He^{\theta_iP_i}\ket{\Psi^{(k)}} \right]_{\theta_i=0} = \bra{\Psi^{(k)}}[H,P_i]\ket{\Psi^{(k)}}.$$

Standard ADAPT-VQE would append only the operator with largest gradient norm. TETRIS instead sorts the pool by descending gradient norm and keeps appending lower-ranked operators so long as each new operator acts on qubits different from those already used in the current TETRIS batch (Anastasiou et al., 2022).

In the notation used in the technical summaries of the method, if $S(P_i)$ denotes the support of operator $P_i$, then two operators may be placed in the same TETRIS iteration only if

$S(P_i)\cap S(P_j)=\varnothing.$

Operationally, the adaptive step is: measure all pool gradients once, sort them, choose the largest-gradient operator, and continue scanning down the sorted list until either all qubits are covered or no nonzero-gradient operator remains with support disjoint from the already chosen set. Only then is the full parameter vector reoptimized. The stopping condition in the original simulations is the standard pool-gradient norm threshold, set to $10^{-7}$, with BFGS used for classical optimization and optimizer gradient-norm tolerance $$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$0 (Anastasiou et al., 2022).

Two clarifications are central. First, TETRIS is not a new objective function; it changes ansatz construction, not energy estimation. Second, no additional gradient measurements are needed inside a single adaptive round beyond those already required to rank the pool.

3. Operator pools and circuit construction

The method is pool-independent in principle, but the original paper mainly studies the qubit pool and the qubit-excitation (QE) pool, precisely because their operators typically have smaller support than fermionic excitations and therefore admit more disjoint packing per iteration (Anastasiou et al., 2022).

For the fermionic pool, the paper uses the usual single- and double-excitation generators. Under Jordan–Wigner, for $$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$1,

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$2

and

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$3

These mapped fermionic operators carry long Jordan–Wigner parity strings and therefore comparatively large support.

The qubit pool decomposes mapped fermionic generators into individual Pauli terms, with examples such as

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$4

The QE pool instead uses qubit raising and lowering operators

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$5

leading to single and double qubit excitations

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$6

Under Jordan–Wigner these become

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$7

and

$$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$8

This pool choice matters because TETRIS gains are driven by disjointness. Smaller-support operators make it easier to populate a single adaptive layer with several nonoverlapping generators. In the reported resource analysis, the circuits were compiled in Qiskit with optimization level 1 and back-to-back gate cancellation, assuming all-to-all qubit connectivity. For qubit-pool operators, Pauli-string exponentials were synthesized through the standard basis-change plus CNOT-staircase construction; for QE-pool operators, the study used the optimized circuits from prior qubit-excitation work (Anastasiou et al., 2022).

4. Resource profile and benchmark behavior

The central empirical claim of the original TETRIS paper is that the method yields denser but significantly shallower ansätze without increasing the number of CNOT gates or variational parameters, while also reducing the number of full pool-gradient measurement rounds because fewer adaptive iterations are required (Anastasiou et al., 2022).

The reported molecular benchmarks use STO-3G, Jordan–Wigner encoding, and the systems linear $$\ket{\Psi(\boldsymbol{\theta})} = \prod_{\ell=1}^{m} e^{\theta_\ell P_\ell}\ket{\Psi_{\mathrm{ref}}},$$9, $$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$0, linear $$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$1, and $$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$2. The advantage grows with system size, which is consistent with the geometric intuition that larger systems admit more simultaneously placeable local operators.

System	Avg. ADAPT/TETRIS depth ratio	Gradient-measurement reduction
$$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$3 (8 qubits)	1.64 (qubit), 1.58 (QE)	53%
$$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$4 (12 qubits)	2.08 (qubit), 1.76 (QE)	59%
$$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$5 (12 qubits)	2.32 (qubit), 2.25 (QE)	66%
$$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$6 (14 qubits)	2.73 (qubit), 2.56 (QE)	70%

These gradient-round reductions were reported without assuming simultaneous measurement grouping. The paper also argues that when the ansatz is dominated by four-qubit double-excitation-like operators, the expected reduction factor in gradient-screening rounds is roughly $$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$7, with $$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$8 the qubit count (Anastasiou et al., 2022).

A second major point is expressivity. In the $$E(\boldsymbol{\theta})=\bra{\Psi(\boldsymbol{\theta})}H\ket{\Psi(\boldsymbol{\theta})}.$$9 example at $P_i$0 Å with the qubit pool and Hartree–Fock reference $P_i$1, standard qubit-ADAPT after one iteration produces

$P_i$2

whereas one TETRIS iteration with two disjoint operators gives

$P_i$3

$P_i$4

The immediate appearance of the quadruple-excitation component $P_i$5 is used in the paper to illustrate faster exploration of the relevant Hilbert space. In the same example, standard ADAPT can stall at an excited eigenstate because the pool gradient vanishes there, whereas TETRIS converges to the exact ground state within 13 iterations (Anastasiou et al., 2022).

The empirical picture in the original study is therefore specific: TETRIS does not merely serialize the same operator list differently; it changes the early expressive content of the ansatz by allowing several disjoint local directions to enter simultaneously.

5. Implementations and later applications

Later work has treated TETRIS-ADAPT-VQE as a concrete depth-aware adaptive strategy rather than a purely conceptual modification. In a $P_i$6 study on IBM hardware and simulators, TETRIS-adaptive VQE was implemented alongside vanilla VQE, ADAPT-VQE, and entanglement forging. In that implementation, vanilla VQE used 1020 two-qubit gates at depth 78, TETRIS used 603 two-qubit gates at depth 22, and their customized ADAPT-VQE used 190 two-qubit gates at depth 15. That study therefore supports TETRIS as a circuit-compression strategy relative to vanilla VQE, but also shows that a particular ADAPT implementation can outperform it on the same instance; the authors explicitly note that their ADAPT procedure used double thresholding and suggest extending that idea to TETRIS in future work (Pandey et al., 2024).

TETRIS has also been tested well beyond the original small-molecule setting. For random all-to-all Hamiltonians, TETRIS-ADAPT-VQE was used to prepare ground states of the quantum Sherrington–Kirkpatrick model and dense and sparse SYK models. The reported fidelities are $P_i$7 for dense and sparse SYK up to $P_i$8 Majorana fermions and $P_i$9 for the quantum SK model up to $k+1$0 sites. The same study reaches a more nuanced scaling conclusion: TETRIS is resource-efficient for SK, but not efficient for dense SYK or moderately sparse SYK, even though state accuracy remains high (Gupta et al., 16 Jun 2026).

The TETRIS growth principle has also been ported to time-dependent variational simulation. AVQDS(T) transfers the same “fill a layer with disjoint unitaries” rule to adaptive quantum dynamics, choosing operators by reduction of the McLachlan distance rather than ground-state energy. In noiseless benchmarks on local spin models, this reduced final circuit depth by more than half relative to the non-TETRIS adaptive baseline, and lowered two-qubit gate count by 22% in TFIM and 26% in MFIM. This is not a VQE result, but it confirms that TETRIS is a general adaptive layer-construction principle rather than a molecule-specific trick (Zhang et al., 2024).

6. Distinctions, complementary methods, and limitations

Several adjacent methods are often conflated with TETRIS-ADAPT-VQE but address different bottlenecks. Operator-pool tiling, for example, learns a compact local operator vocabulary on a small lattice instance and replicates it to larger systems; it changes pool construction rather than the ansatz-growth rule. The tiling paper explicitly frames this as complementary to TETRIS: tiling reduces pool size and gradient-screening burden, whereas TETRIS reduces circuit depth and serial growth by packing nonoverlapping selected operators into a common adaptive round (Dyke et al., 2022).

Likewise, TETRIS should not be confused with the separate compiler framework “Tetris,” which targets Pauli-string synthesis and hardware mapping to reduce two-qubit gate counts in VQA circuits. That compiler is technically relevant to chemistry-style ansätze compiled into Pauli exponentials, but it is not an ADAPT ansatz-selection rule and does not study TETRIS-ADAPT-VQE directly (Jin et al., 2023).

Other later ADAPT developments attack bottlenecks that remain present inside TETRIS. Gradient measurement is one such case. A commuting-observable strategy for ADAPT pool gradients reduces the observable-group scaling for hardware-efficient pools from naive $k+1$1 counting to $k+1$2 commuting sets, with worst-case full-pool gradient measurement only $k+1$3 times as expensive as a naive VQE iteration. Because TETRIS still depends on full-pool ranking information, this type of measurement reduction is directly complementary rather than competitive (Anastasiou et al., 2023).

Insertion-position sensitivity is another orthogonal issue. Work on gradient troughs shows that append-only ADAPT can stagnate because the usefulness of an operator depends on where it is inserted in a noncommuting ansatz. That line of work is not TETRIS—it does not batch disjoint operators—but it reinforces a general lesson already implicit in TETRIS: ansatz placement is a first-class design variable, not merely a compilation afterthought (Stadelmann et al., 31 Dec 2025).

At the simulation layer, sparse-Taylor state-evolution methods for ADAPT-style workflows provide yet another complement. A deterministic fifth-order Taylor update implemented as chained sparse matrix-vector multiplications is explicitly described as compatible with operator-pool-based adaptive variants, but it is not a TETRIS-specific selection strategy. A plausible implication is that such a backend can accelerate the classical state-update portion of TETRIS-style simulations when sparse operator representations are used (Dipojono, 29 Jun 2026).

The original TETRIS study also has clear limitations. Its resource comparisons assume all-to-all connectivity, its strongest gains appear with relatively local qubit or QE pools, and its evidence is primarily numerical rather than theorem-driven: the paper does not prove a general convergence or optimality result for batched disjoint-support growth. It is therefore best understood as a concrete adaptive ansatz-construction rule whose benefits are empirically strong in the tested settings, especially as system size increases, but whose ultimate performance remains pool-, mapping-, and hardware-dependent (Anastasiou et al., 2022).