Fraxis: Free-Axis Selection for PQCs

• Fraxis is a sequential, gradient-free optimizer for parameterized quantum circuits that updates one single-qubit gate at a time via adaptive free-axis selection.
• It employs local eigendecomposition and parameter-shift evaluations to solve low-dimensional subproblems, ensuring predictable measurement costs and improved convergence.
• Extensions like gate freezing and two-gate variants enhance efficiency by reallocating measurement budgets and reducing optimization errors on benchmark Hamiltonians.

Searching arXiv for papers on Fraxis / Free-Axis Selection to ground the article in published work. arxiv_search(query="Fraxis OR \"Free-Axis Selection\" parameterized quantum circuit", max_results=10, sort_by="submittedDate") Fraxis, short for Free-Axis Selection, is a sequential, gate-by-gate, gradient-free optimizer for parameterized quantum circuits (PQCs) used in variational quantum algorithms on noisy intermediate-scale quantum devices. In the cited literature, it is presented as a single-qubit local optimizer that updates one gate at a time while holding the rest of the circuit fixed, with later work extending it through gate freezing, hybrid switching schemes, and two-gate variants. Across these formulations, Fraxis occupies an intermediate position between the lower-cost Rotosolve update and the more expressive but more measurement-intensive Free-Quaternion Selection (FQS) update (Pankkonen et al., 10 Jul 2025, Pankkonen et al., 9 Oct 2025, Pankkonen, 26 Mar 2026).

1. Formal setting and optimization target

Fraxis is defined on an $n$-qubit PQC of the form

$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$

where each layer is written as

$$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$

with $W_l$ an entanglement block and $H_d$ Hermitian single-qubit generators, typically Pauli operators. The global objective is the expectation value

$$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$

for an observable $M$ (Pankkonen et al., 10 Jul 2025).

The optimizer is explicitly local. When the $d$-th gate is targeted, the circuit is partitioned into fixed “before” and “after” blocks, typically denoted $V_2$ and $V_1$, so that all gates except the target are absorbed into an effective state and an effective observable: $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$0 The local optimization step is then performed on a single-qubit subproblem defined by these dressed objects rather than on the full circuit (Pankkonen et al., 10 Jul 2025).

This single-gate structure is central to the method’s role in VQA optimization. It yields closed-form or low-dimensional subproblems, avoids explicit gradient estimation, and makes the measurement budget per gate update predictable. At the same time, it means that Fraxis is not a global optimizer in one shot; it is an iterative coordinate-style procedure whose behavior depends on repeated sweeps over all single-qubit gates.

2. Original single-gate Fraxis formulation

In the original formulation recalled in the gate-freezing and two-gate extension papers, the $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$1-th gate is parameterized as a rotation by a fixed angle $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$2 about an optimizable unit axis $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$3, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$4: $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$5 The local cost becomes

$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$6

Fraxis then minimizes over the axis $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$7 subject to the unit-norm constraint (Pankkonen et al., 10 Jul 2025).

The constrained problem is written through the Lagrangian

$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$8

Stationarity, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$9, leads to a symmetric $$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$0 eigenvalue problem

$$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$1

The construction of $$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$2 requires six expectation values: $$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$3 together with

$$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$4

and the analogous $$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$5 and $$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$6. The optimal axis $$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$7 is the eigenvector of $$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$8 corresponding to its smallest eigenvalue (Pankkonen et al., 10 Jul 2025).

An equivalent notation used in the two-gate extension paper writes the local objective as a quadratic form

$$U_l(\vec\theta_l)=W_l\;\bigotimes_{k=0}^{n-1}\exp\!\left(-i\theta_{n(l-1)+k}H_{n(l-1)+k}/2\right),$$9

where $W_l$0 is a real symmetric $W_l$1 matrix with entries

$W_l$2

The diagonal terms $W_l$3 and off-diagonal terms are reconstructed from six circuit evaluations, after which one diagonalizes $W_l$4 and replaces $W_l$5 by $W_l$6. The classical cost of the $W_l$7 eigendecomposition is $W_l$8, and the quantum cost is 6 circuit evaluations per gate (Pankkonen, 26 Mar 2026).

Operationally, one full Fraxis pass initializes $W_l$9 randomly on the unit sphere for all $H_d$0, sweeps sequentially over all $H_d$1 gates, constructs the local dressed problem for each gate, measures the six required quantities, solves the eigenvalue problem, and updates the gate axis. The sweep is repeated until a convergence criterion is met (Pankkonen et al., 10 Jul 2025).

3. Axis-selection formulation in hybrid-optimizer studies

The hybrid-optimizer literature presents Fraxis through a second local formulation. There, the local subcircuit for the $H_d$2-th gate is written as

$H_d$3

with all other gates fixed, and one shows that the local cost has the form

$H_d$4

for a fixed candidate generator $H_d$5. The coefficients are obtained through parameter-shift evaluations: $H_d$6 and

$H_d$7

The minimizing angle is

$H_d$8

with minimum value

$H_d$9

Fraxis is then described as looping over the three Pauli axes, computing $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$0, finding $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$1, choosing

$$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$2

and updating both the axis and the angle of the gate (Pankkonen et al., 9 Oct 2025).

In this presentation, one full pass over a circuit with $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$3 layers and $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$4 qubits uses 6 circuit evaluations per gate per iteration, for a total budget of at most $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$5 evaluations per pass. The same source states that Fraxis is strictly more expressive than Rotosolve, which picks a fixed axis a priori, and cheaper than FQS, which optimizes a full four-parameter quaternion (Pankkonen et al., 9 Oct 2025).

The coexistence of this description with the $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$6-rotation eigenproblem formulation indicates that the cited literature uses the name “Fraxis” across related sequential free-axis update rules. A plausible implication is that the term is best understood operationally—as a family of local axis-adaptive single-qubit updates—rather than as a single invariant parametrization.

4. Gate freezing and adaptive update scheduling

The gate-freezing extension modifies Fraxis by exploiting the observation that some gate parameters may change only negligibly from one sweep to the next. The stated rationale is that future effort is then better spent on more “active” gates. For Fraxis, the tracked quantity is the axis update $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$7, and the distance is measured on the unit sphere with the identification $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$8, reflecting global phase invariance for $$C(\vec\theta)=\langle M\rangle=\mathrm{Tr}[M\,U(\vec\theta)\,\rho_0\,U(\vec\theta)^\dagger], \qquad \rho_0=|0^n\rangle\langle 0^n|,$$9-rotations: $M$0 Given a threshold $M$1, a gate is declared “well-optimized” whenever

$M$2

and is then frozen for the next $M$3 iterations, meaning that its update is skipped during those passes (Pankkonen et al., 10 Jul 2025).

The extension also includes an incremental freeze length. Instead of a uniform fixed $M$4, each gate $M$5 may be assigned its own freeze counter $M$6, initially $M$7, with the rule $M$8 each time the gate is frozen. This increasingly penalizes gates that repeatedly fail to move (Pankkonen et al., 10 Jul 2025).

The modified pseudocode introduces a per-gate freeze state $M$9, the number of remaining iterations for which the gate is frozen. If $d$0, one decrements $d$1 and skips the update. Otherwise, one performs the usual Fraxis local measurement and eigensolve, computes $d$2, and if $d$3 sets $d$4 and then increments $d$5. The process continues until the total gate-update count reaches the baseline Fraxis budget or convergence occurs (Pankkonen et al., 10 Jul 2025).

The practical recommendations are specific. Best performance is reported for a small threshold $d$6 and short freeze length $d$7–5, or with adaptive $d$8. The stated interpretation is that freezing reallocates measurement budget to “active” gates, improving sample-efficiency. The stated limitations are equally specific: $d$9 and the initial $V_2$0 must be tuned per problem, and all tests were noiseless, so performance on real noisy hardware remains to be assessed (Pankkonen et al., 10 Jul 2025).

5. Benchmark behavior on spin and fermionic Hamiltonians

The gate-freezing study evaluates Fraxis and its freezing variants in noiseless simulations using PennyLane 0.40 on classical hardware. Two benchmark Hamiltonians are reported: a 1D Heisenberg model and a $V_2$1 Fermi–Hubbard model (Pankkonen et al., 10 Jul 2025).

For the 1D Heisenberg model, the Hamiltonian is

$V_2$2

with $V_2$3 qubits and periodic boundary, and ground-state energy $V_2$4. The PQC ansatz uses 5 layers of single-qubit Fraxis rotations and an entangling CZ network. The total gate-update budget per run is $V_2$5, with twenty independent random starts. The reported median final energies are summarized below.

Setting	Baseline Fraxis	Best reported median
1D Heisenberg, $V_2$6	$V_2$7	$V_2$8

At fixed $V_2$9, the medians are $V_1$0 for $V_1$1, $V_1$2 for $V_1$3, and $V_1$4 for $V_1$5, compared with $V_1$6 for the base method. At fixed $V_1$7, they are $V_1$8, $V_1$9, and $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$00, respectively. For incremental $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$01, they are $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$02, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$03, and $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$04. The interquartile range in the box plot shrank by approximately $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$05 under $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$06 versus the base method, and gate freezing cut the energy gap to $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$07 by approximately $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$08 on average (Pankkonen et al., 10 Jul 2025).

For the $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$09 Fermi–Hubbard model, the system is mapped to 6 qubits via Jordan–Wigner, with tunneling $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$10 and ground energy $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$11. Fraxis uses 5 layers and 30 passes. The reported median final energies over 20 runs are summarized below.

Setting	Baseline Fraxis	Best reported median
$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$12 Fermi–Hubbard, 6 qubits	$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$13	$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$14

For $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$15, the medians are $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$16 at $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$17, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$18 at $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$19, and $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$20 at $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$21, versus $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$22 for the base method. For $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$23, they are $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$24, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$25, and $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$26. For incremental $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$27, they are $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$28, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$29, and $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$30. The reported summary is that gate freezing reduces median energy error by approximately $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$31 versus the base method (Pankkonen et al., 10 Jul 2025).

The same study tracks final $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$32 values through freeze-count heat maps. Gates in the last PQC layer, especially the “middle” qubit, consistently accrued the largest freeze counts, indicating that they stabilized earliest, while early-layer gates tended never to freeze. This suggests a nonuniform distribution of local optimization difficulty across circuit depth (Pankkonen et al., 10 Jul 2025).

6. Hybrids, shot noise, and scaling behavior

Fraxis is also studied as a standalone method and as a component in hybrid switching schemes. In the hybrid study, all experiments were carried out in PennyLane on three problem classes: the 1D Heisenberg model for $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$33 qubits, the 2D Fermi–Hubbard model on $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$34 mapped to 6 qubits, and random 4-qubit state-fidelity maximization (Pankkonen et al., 9 Oct 2025).

Two switching strategies are defined. In Algorithm 1, termed the early-stopping switch, one starts with $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$35 Rotosolve, computes $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$36 after each gate update, increments a patience counter when $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$37, and switches globally to $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$38 FQS when the patience reaches $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$39. In Algorithm 2, termed the cost-average switch, one again starts with Rotosolve, maintains a sliding window of the last $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$40 values of the cost, and switches when $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$41. The study explicitly notes that one may replace $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$42 FQS by $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$43 Fraxis, yielding a Roto-Fraxis hybrid (Pankkonen et al., 9 Oct 2025).

The reported empirical role of Fraxis is consistent across several settings. On the Heisenberg benchmark with 10 qubits and 15 layers, standalone Fraxis converges markedly faster than Rotosolve but slower than FQS. Under shot noise in the range 2048 to 8192 shots, Fraxis retains its advantage over FQS, which suffers from heavier measurement noise due to 10 evaluations, and generally outperforms Rotosolve. Early-stopping or cost-average hybrids with Rotosolve$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$44FQS beat all standalone methods, with Fraxis identified as the next best standalone method (Pankkonen et al., 9 Oct 2025).

On the $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$45 Fermi–Hubbard benchmark with 6 qubits and 5 layers, Fraxis again interpolates between Rotosolve and FQS in convergence speed. Under low-shot budgets of 2048 shots, Fraxis often outperforms FQS because its 6 evaluations yield lower statistical noise than the 10 evaluations required by FQS. In the scalability study for $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$46 qubits with $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$47 layers, Fraxis alone scales much better than Rotosolve but less well than the hybrid methods. On random 4-qubit fidelity maximization, FQS dominates, Fraxis comes second, and the hybrids trail (Pankkonen et al., 9 Oct 2025).

The practical guidelines attached to this study are quantitative. Fraxis costs 6 circuit calls per gate per pass, compared with 3 for Rotosolve and 10 for FQS. Suggested hybrid parameters are $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$48 with patience $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$49 for early stopping, and window $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$50 with $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$51 for the cost-average switch. The recommended shot range is 2 000–8 000 shots per expectation, with the specific claim that Fraxis is more noise-robust than FQS due to fewer evaluations (Pankkonen et al., 9 Oct 2025).

7. Two-gate generalization and outstanding technical questions

The two-gate extension, Two-Gate Fraxis (TGF), generalizes the single-gate local step by optimizing two parameterized single-qubit gates simultaneously. If the two gates are $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$52 and $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$53, with the fixed block between them denoted $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$54, the local cost is

$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$55

Each gate is parameterized by a unit quaternion $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$56, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$57, and after expansion the cost becomes an exact quartic polynomial in $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$58 and $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$59. The coefficients are reconstructed from circuit evaluations, and the constrained classical problem

$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$60

is solved by an off-the-shelf nonlinear optimizer; the implementation reported in the paper uses SLSQP (Pankkonen, 26 Mar 2026).

The paper distinguishes four gate-pairing strategies for a circuit with $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$61 single-qubit gates: linear, random, opposite, and half-shifted. It also gives the measurement overhead explicitly. Fraxis costs 6 circuit evaluations per gate update, TGF costs 36 per gate update, which is 18 per gate, and the two-gate quaternion variant TGFQS costs 100 per gate update, or 50 per gate. The stated trade-off is between stronger local optimization and increased measurement overhead (Pankkonen, 26 Mar 2026).

Benchmark	Best TGF/TGFQS improvement	Best pairing
Fermi–Hubbard $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$62, 4 qubits	$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$63 / $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$64 drop in error	random
TFIM, 12 qubits, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$65	$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$66 / $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$67 drop	half-shifted
LiH, 12 qubits, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$68	$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$69 / $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$70 drop	random / opposite
BeH$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$71, 14 qubits, $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$72	$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$73 / $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$74 drop	linear
Fidelity maximization, 6 qubits	$$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$75 / $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$76 further drop	half-shifted / random

The reported conclusion is that TGF and TGFQS frequently achieve a lower final relative error to the ground-state energy or infidelity than their single-gate counterparts, and that the random and half-shifted pairing strategies perform best in many tested settings. In additional finite-shot experiments on Fermi–Hubbard and transverse-field Ising Hamiltonians, the best pairing strategies retain their advantage across the tested shot counts in shallow circuits. The gain is stated to be most pronounced when the circuit has moderate depth and the shot noise is not too large, with the summary recommendation that TGF is often attractive when 20–50 measurements per gate update are affordable and circuit depths are shallow, roughly $$U(\vec\theta)=U_L(\vec\theta_L)\cdots U_1(\vec\theta_1),$$77–5 on up to approximately 15 qubits (Pankkonen, 26 Mar 2026).

Several open technical directions are explicit in the current Fraxis literature. The gate-freezing work proposes extending freezing to multiqubit blocks via unitary-norm metrics, co-designing ansatz structure informed by freeze heat maps, and validating under realistic noise models. Because all gate-freezing tests were noiseless, assessment on real noisy hardware remains unresolved (Pankkonen et al., 10 Jul 2025). Taken together, these directions place Fraxis within an active line of research on analytic local updates, adaptive measurement allocation, and progressively larger local subproblems for PQC optimization.