⚛️ quantum physics

Two-Gate Extensions of Free Axis and Free Quaternion Selection for Sequential Optimization of Parameterized Quantum Circuits

This paper introduces Two-Gate Fraxis (TGF) and Two-Gate FQS (TGFQS), which extend existing sequential optimizers by simultaneously updating two single-qubit gates via an exact quartic cost function, demonstrating improved convergence to ground states and reduced infidelity in various quantum tasks despite increased measurement overhead.

Original authors: Joona V. Pankkonen

Published 2026-03-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Joona V. Pankkonen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to tune a massive, complex radio to find a single, perfect station in a sea of static. This radio has hundreds of dials (quantum gates), and your goal is to turn them just right so the music (the solution to a problem) comes through clearly.

This is the challenge of Variational Quantum Algorithms (VQAs). You have a quantum circuit (the radio) with many adjustable knobs. You need to find the perfect combination of settings to get the best result, like finding the lowest energy state of a molecule or the ground state of a magnetic material.

The Old Way: Tuning One Knob at a Time

For a long time, the standard method to tune these radios was sequential optimization. Think of this like a team of mechanics where only one mechanic is allowed to touch one single dial at a time.

Two popular methods for this were called Fraxis and FQS.

How they worked: The mechanic would freeze every other dial, fiddle with just one, measure the result, and then move to the next dial.
The Math: Because they were only changing one thing at a time, the math was relatively simple (like solving a quadratic equation). They could find the perfect setting for that one dial very quickly by looking at a simple map.
The Problem: If you have 100 dials, you have to visit them one by one, over and over again. It's slow, and sometimes you get stuck in a "local valley"—a spot that looks like the bottom of a hill, but isn't the actual bottom of the mountain.

The New Idea: Two Mechanics, Two Knobs

The authors of this paper, Joona Pankkonen, proposed a bold new strategy: What if two mechanics could tune two dials at the same time?

They introduced two new methods: TGF (Two-Gate Fraxis) and TGFQS (Two-Gate FQS).

Instead of looking at one dial in isolation, they look at pairs of dials.

The Analogy: Imagine you are trying to balance a tray with two heavy plates on it. If you only adjust the left hand, the tray might tip. If you only adjust the right hand, it might tip the other way. But if you coordinate your hands to move both simultaneously, you can find a perfect balance much faster.
The Math: This is harder. When you move two dials at once, the relationship between the settings and the result isn't a simple curve anymore; it's a complex, bumpy 4D landscape (a "quartic" function). You can't just look at a simple map; you need a sophisticated GPS (a classical computer optimizer) to navigate this bumpy terrain.

The Secret Sauce: How to Pair the Dials

The researchers realized that just picking any two dials to tune together isn't enough. You have to be smart about which dials you pair. They tested four strategies:

Linear: Pairing neighbors (Dial 1 with 2, 3 with 4).
Opposite: Pairing the first with the last (Dial 1 with 100).
Random: Picking two dials at random.
Half-Shifted: Pairing Dial 1 with Dial 51, Dial 2 with 52, etc.

The Surprise: The "Random" and "Half-Shifted" strategies worked the best! It turns out that pairing dials that are far apart or randomly selected helps the system escape those "local valleys" and find the true global optimum much faster.

The Trade-off: More Work for Better Results

There is a catch. Tuning two dials at once requires more measurements.

Old Way: To tune one dial, you needed about 6 to 10 measurements.
New Way: To tune a pair of dials, you need about 36 to 100 measurements.

It's like the new method requires the mechanics to take more photos of the dial settings before deciding how to turn them. However, the paper shows that this extra effort pays off. Even though they take more measurements per step, they reach the perfect solution with far fewer total steps. In many tests (like simulating molecules or magnetic materials), the new method found a much more accurate answer than the old method, sometimes reducing the error by nearly 100%.

Real-World Tests

The team tested this on:

Fermi-Hubbard Model: Simulating how electrons hop around in a grid (like a game of musical chairs for electrons).
Ising Model: Simulating magnetic materials.
Molecules: Calculating the energy of Lithium Hydride and Beryllium Hydride (important for chemistry).
State Preparation: Trying to create a specific quantum state from scratch.

In almost every case, the "Two-Gate" methods (TGF and TGFQS) with the right pairing strategy (Random or Half-Shifted) won the race. They found the "perfect station" faster and with less static (error) than the old "one-dial-at-a-time" methods.

The Bottom Line

This paper is about upgrading the "tuning" process for quantum computers. By allowing two parameters to be optimized simultaneously and being clever about which ones to pair, we can solve complex quantum problems more accurately. It's a bit like realizing that while tuning a guitar one string at a time is okay, tuning two strings together in harmony gets you the right chord much faster, even if it takes a few extra minutes to figure out the perfect twist for both.

1. Problem Statement

Variational Quantum Algorithms (VQAs) rely on Parameterized Quantum Circuits (PQCs) optimized via hybrid quantum-classical loops. A major bottleneck in PQC optimization is the Barren Plateau problem, where gradients vanish exponentially as qubit count increases, making training difficult. Additionally, the NP-hard nature of optimizing large circuits and the high measurement overhead (shot noise) pose significant challenges.

Existing sequential optimizers, such as Free Axis Selection (Fraxis) and Free Quaternion Selection (FQS), address these issues by optimizing one single-qubit gate at a time. They construct a quadratic local cost function based on circuit evaluations and solve it analytically via matrix diagonalization. However, these methods treat gates independently, potentially missing correlations between parameters that could accelerate convergence or escape local minima.

2. Methodology

The authors propose Two-Gate Fraxis (TGF) and Two-Gate FQS (TGFQS), which extend the sequential optimization framework to update two parameterized single-qubit gates simultaneously.

Core Mathematical Formulation

Quartic Cost Function: Unlike the quadratic cost functions of single-gate optimizers, optimizing two gates ( $R_d$ $R_{d}$ and $R_k$ $R_{k}$ ) simultaneously results in a quartic (fourth-order) local cost function.
- For FQS, gates are parameterized by unit quaternions ( $q \in \mathbb{R}^4$ ). The cost function becomes a quartic polynomial in terms of the components of two quaternions ( $q_d, q_k$ ).
- For Fraxis, the rotation angle is fixed ( $\theta = \pi$ ), reducing the problem to optimizing two rotation axes, which is a special case of TGFQS.
Exact Coefficient Construction: The authors derive an exact method to construct this quartic cost function using circuit evaluations. They define a set of coefficients ( $A_{\mu\nu\alpha\beta}$ $A_{μν α β}$ ) derived from expectation values of the Hamiltonian. Using polarization identities (similar to those used in single-gate methods), they express these coefficients via linear combinations of circuit evaluations with specific operator insertions.
- TGFQS requires 100 circuit evaluations to update a pair of gates (50 per gate on average).
- TGF requires 36 circuit evaluations to update a pair (18 per gate on average).
Classical Optimization: Since the cost function is quartic, an analytic closed-form solution (like eigenvalue decomposition) is generally unavailable. The authors employ constrained classical optimizers (specifically SLSQP) to find the optimal quaternions subject to the unit norm constraint ( $|q|=1$ ).

Gate Pairing Strategies

A critical component of the method is how gates are paired for simultaneous optimization. The paper investigates four strategies:

Linear: Adjacent gates $(1,2), (3,4), \dots$
Random: Randomly permuted pairs.
Opposite: Gates at opposite ends of the circuit $(1, D), (2, D-1), \dots$
Half-Shifted: Gates separated by half the circuit depth $(1, D/2+1), \dots$

3. Key Contributions

Algorithmic Extension: The first extension of Fraxis and FQS to simultaneous two-gate updates, transforming the local optimization problem from quadratic to quartic.
Exact Cost Function Derivation: A rigorous derivation of the quartic cost function coefficients using only expectation values, avoiding the need for full state tomography.
Pairing Strategy Analysis: A systematic study demonstrating that the choice of gate pairing significantly impacts convergence, with Random and Half-Shifted strategies generally outperforming Linear and Opposite pairings.
Empirical Validation: Comprehensive benchmarking across diverse tasks (spin models, molecular Hamiltonians, and state preparation) under both noiseless and finite-shot conditions.

4. Results

The methods were tested on the Fermi-Hubbard model, Transverse-Field Ising Model (TFIM), LiH and BeH2 molecular Hamiltonians, and a Quantum State Preparation task.

Performance Improvement: TGF and TGFQS consistently achieved lower final relative errors (closer to the ground state energy) and lower infidelity compared to their single-gate counterparts (Fraxis and FQS).
- In the Fermi-Hubbard model, TGF/TGFQS with random pairing reduced relative error by orders of magnitude compared to Fraxis.
- In molecular Hamiltonians (LiH, BeH2), the two-gate methods achieved significantly better convergence despite the higher measurement cost.
Gate Pairing Impact:
- Random and Half-Shifted pairings were the most effective strategies across most benchmarks.
- Linear and Opposite pairings often performed similarly to or only marginally better than the single-gate baseline, suggesting that the spatial correlation of gates matters less than the diversity of the pairing.
Finite-Shot Robustness: In experiments simulating finite measurement accuracy (4k, 8k, 16k shots), the advantage of TGF/TGFQS (specifically with Random/Half-Shifted pairings) was retained, particularly in shallow circuits ( $L=2$ ). However, TGFQS showed higher sensitivity to shot noise than TGF.
Comparison with Gradient-Based Methods: When compared to the Adam optimizer (using parameter-shift rules) adjusted for equivalent circuit evaluations, TGF and TGFQS with optimal pairing strategies achieved superior convergence and lower final errors.

5. Significance and Trade-offs

Trade-off: The primary cost of TGF/TGFQS is the measurement overhead. Updating two gates requires significantly more circuit evaluations (18x for TGF, 50x for TGFQS per gate) compared to single-gate updates (6x for Fraxis, 10x for FQS).
Mitigation: The authors note that the required circuit evaluations are mutually independent, meaning they can be executed in parallel across multiple quantum devices or shots, potentially mitigating the wall-clock time penalty.
Implication: The results suggest that local multi-parameter optimization is a powerful strategy for VQAs. By capturing correlations between gates, these methods can navigate the optimization landscape more effectively than sequential single-gate updates, offering a viable path to mitigate barren plateaus and improve ground state estimation in NISQ devices, provided the measurement overhead can be managed.

Conclusion

The paper establishes that extending sequential optimizers to update two gates simultaneously yields substantial performance gains in terms of solution quality. While the measurement cost is higher, the ability to find better optima with fewer iterations (in terms of gate updates) and the potential for parallel execution make TGF and TGFQS promising candidates for optimizing complex parameterized quantum circuits. Future work may explore extending this to $m$ -gate updates or developing approximation strategies to reduce the exponential scaling of measurement terms.