--- title: 'Fraxis: Free-Axis Selection for PQCs' url: https://www.emergentmind.com/topics/fraxis type: topic --- # Fraxis: Free-Axis Selection for PQCs Searching arXiv for recent 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 [2507.07742, 2510.08142, 2603.25876]. ## 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 \(M\) [2507.07742]. 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: \[ \rho \equiv V_2\,\rho_0\,V_2^\dagger, \qquad M \equiv V_1^\dagger\,M\,V_1. \] The local optimization step is then performed on a single-qubit subproblem defined by these dressed objects rather than on the full circuit [2507.07742]. 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 \(d\)-th gate is parameterized as a rotation by a fixed angle \(\pi\) about an optimizable unit axis \(\hat n_d\in\mathbb R^3\), \(\|\hat n_d\|=1\): \[ R_d(\pi,\hat n_d)=\exp\!\left[-i(\pi/2)(\hat n_d\cdot \sigma)\right], \qquad \sigma=(X,Y,Z). \] The local cost becomes \[ \langle M\rangle_{\hat n_d} = \mathrm{Tr}\!\left[M\,R_d(\pi,\hat n_d)\,\rho\,R_d(\pi,\hat n_d)^\dagger\right]. \] Fraxis then minimizes over the axis \(\hat n\) subject to the unit-norm constraint [2507.07742]. The constrained problem is written through the Lagrangian \[ f(\hat n,\lambda)=\langle M\rangle_{\hat n}-\lambda(n_x^2+n_y^2+n_z^2-1). \] Stationarity, \(\partial f/\partial n_i=0\), leads to a symmetric \(3\times 3\) eigenvalue problem \[ A\hat n = 2\lambda \hat n. \] The construction of \(A\) requires six expectation values: \[ r_x=\mathrm{Tr}[M X\rho X],\quad r_y=\mathrm{Tr}[M Y\rho Y],\quad r_z=\mathrm{Tr}[M Z\rho Z], \] together with \[ r_{x+y}=\mathrm{Tr}\!\left[M\frac{X+Y}{\sqrt2}\rho\frac{X+Y}{\sqrt2}\right], \] and the analogous \(r_{x+z}\) and \(r_{y+z}\). The optimal axis \(\hat n^\ast\) is the eigenvector of \(A\) corresponding to its smallest eigenvalue [2507.07742]. An equivalent notation used in the two-gate extension paper writes the local objective as a quadratic form \[ \langle M\rangle(\hat n)=\hat n^\top \tilde S\,\hat n, \] where \(\tilde S\) is a real symmetric \(3\times 3\) matrix with entries \[ \tilde S_{ab}=\mathrm{Tr}[M'\sigma_a'\rho'\sigma_b']. \] The diagonal terms \(\tilde S_{aa}\) and off-diagonal terms are reconstructed from six circuit evaluations, after which one diagonalizes \(\tilde S\) and replaces \(R_{\hat n}(\pi)\) by \(R_{\hat n^\ast}(\pi)\). The classical cost of the \(3\times 3\) eigendecomposition is \(O(1)\), and the quantum cost is 6 circuit evaluations per gate [2603.25876]. Operationally, one full Fraxis pass initializes \(\hat n_d\) randomly on the unit sphere for all \(d\), sweeps sequentially over all \(D=L\cdot n\) 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 [2507.07742]. ## 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 \(d\)-th gate is written as \[ V_1\,e^{-i\theta H/2}\,V_2, \] with all other gates fixed, and one shows that the local cost has the form \[ C(\theta)=A\cos\theta + B\sin\theta + C_0, \] for a fixed candidate generator \(H\in\{X,Y,Z\}\). The coefficients are obtained through parameter-shift evaluations: \[ A = [C(0)-C(\pi)]/2,\qquad C_0=[C(0)+C(\pi)]/2, \] and \[ B=[C(\pi/2)-C(-\pi/2)]/2. \] The minimizing angle is \[ \theta^\ast=\mathrm{atan2}(B,A)+\pi, \] with minimum value \[ C_{\min}=C_0-\sqrt{A^2+B^2}. \] Fraxis is then described as looping over the three Pauli axes, computing \((A_H,B_H,C_{0,H})\), finding \((\theta_H^\ast,C_{H,\min})\), choosing \[ H^\star=\arg\min_{H\in\{X,Y,Z\}} C_{H,\min}, \] and updating both the axis and the angle of the gate [2510.08142]. In this presentation, one full pass over a circuit with \(L\) layers and \(n\) qubits uses 6 circuit evaluations per gate per iteration, for a total budget of at most \(6Ln\) 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 [2510.08142]. The coexistence of this description with the \(\pi\)-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 \(\hat n_{d,\mathrm{prev}}\to \hat n_{d,\mathrm{new}}\), and the distance is measured on the unit sphere with the identification \(\hat n\leftrightarrow -\hat n\), reflecting global phase invariance for \(\pi\)-rotations: \[ D(\hat n_{\mathrm{prev}},\hat n_{\mathrm{new}}) = \min\Bigl\{ \arccos(\hat n_{\mathrm{prev}}\!\cdot\!\hat n_{\mathrm{new}}), \, \pi-\arccos(\hat n_{\mathrm{prev}}\!\cdot\!\hat n_{\mathrm{new}}) \Bigr\}. \] Given a threshold \(T\in[0,\pi]\), a gate is declared “well-optimized” whenever \[ D(\hat n_{d,\mathrm{prev}},\hat n_{d,\mathrm{new}})0\), one decrements \(F_d\) and skips the update. Otherwise, one performs the usual Fraxis local measurement and eigensolve, computes \(\Delta=D(\hat n_{d,\mathrm{prev}},\hat n_{d,\mathrm{new}})\), and if \(\Delta