⚛️ quantum physics

Optimised T counts and active volume estimates for high- and low- level arithmetic subroutines

This paper presents optimized T-counts and active volume estimates for high- and low-level arithmetic subroutines on surface code quantum computers, demonstrating that circuit structure significantly impacts active volume and proposing a method using oriented ZX diagrams to further reduce spacetime overhead.

Original authors: Sam Heavey

Published 2026-09-09

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CC BY 4.0

Original authors: Sam Heavey

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

Quantum computers promise to solve problems that would take classical machines thousands of years, but they face a fundamental hurdle: they are incredibly fragile. The slightest disturbance from the environment can corrupt the calculation, so scientists must use error correction to keep the data safe. One of the most promising methods for this is called the surface code, which treats a single piece of information not as one tiny particle, but as a large, resilient patch made of many physical particles working together. To run a complex program, these patches must be arranged and connected in a specific way. For years, the standard approach to building these machines assumed a rigid, grid-like structure where the patches sit in fixed positions. This design forces many of the patches to sit idle, waiting for their turn to participate, which wastes a tremendous amount of time and space. A newer, more flexible approach known as the active volume architecture aims to fix this by allowing the computer to move data around freely, ensuring that every part of the machine is working whenever possible, much like a busy factory floor where no worker ever stands still.

In this work, researchers at Imperial College London set out to see how much more efficient this active volume approach could be if they redesigned the basic building blocks of quantum math. Just as a human calculator needs to know how to add, subtract, multiply, and divide, a quantum computer needs specialized routines to perform these same tasks. The team took the most advanced designs for these arithmetic routines and re-engineered them specifically for the active volume architecture. They focused on two main goals: reducing the number of "magic" resources required to make the calculations work and minimizing the total space the calculation occupies while it runs. By using a visual method that maps out the connections between quantum bits, they were able to strip away unnecessary steps and rearrange the flow of data to eliminate idle time.

The results show that this new way of thinking about circuit design leads to dramatic savings. For the basic task of multiplying two numbers, the researchers found a design that uses significantly fewer resources than previous methods, cutting the leading cost factor by more than half. They applied similar improvements to more complex functions, such as calculating square roots, trigonometric values like sine, and logarithms. In every case, the new designs required fewer of the expensive, error-prone operations that slow down quantum computers. One of the most surprising discoveries was that the physical arrangement of the gates mattered just as much as the number of gates used. Two circuits that performed the exact same mathematical task with the same number of steps could have very different costs depending on how those steps were ordered. This means that simply counting the parts of a quantum program is not enough; the structure of the program itself determines how efficiently it can run.

The researchers also demonstrated that these savings are not just theoretical. They provided a clear, step-by-step method for calculating the true cost of any quantum routine in this new architecture, moving beyond simple gate counts to measure the actual "active volume" of the computation. This volume represents the amount of space and time the computer is genuinely busy doing work, excluding all the waiting periods that plague older designs. By optimizing the low-level math routines that high-level algorithms depend on, the team has created a catalog of more efficient tools for future quantum software. While these improvements were verified through simulation rather than on a physical machine, the logic is sound and the potential impact is substantial. If these optimized routines are adopted, it could mean that the same quantum computer can solve difficult problems much faster, or that a smaller, cheaper machine could achieve the same results as a much larger one. The work suggests that the path to practical quantum computing lies not just in building better hardware, but in reimagining how we organize the software that runs on it.

Title: Optimised T counts and active volume estimates for high- and low-level arithmetic subroutines

Author: Sam Heavey (Imperial College London)

Problem Statement

Surface code-based quantum computers are a leading candidate for fault-tolerant quantum computing. However, traditional "baseline" architectures, which rely on 2-D arrays of locally connected qubits, suffer from significant inefficiencies. A large portion of the spacetime volume in these computations is "idle," corresponding to logical qubits that sit unused during specific operations. This idle volume increases the overall resource cost (spacetime volume) without contributing to the logical computation.

While "active volume" (AV) architectures aim to eliminate this idle volume by utilizing long-range connectivity to parallelize operations, there is a lack of a comprehensive catalogue of optimised arithmetic subroutines with accurate active volume estimates. Existing literature often focuses on T-count reductions in baseline architectures without fully accounting for how circuit structure impacts active volume, particularly when state injection and reactive measurements are involved.

Methodology

The paper proposes a methodology to derive and optimise the active volume and T-counts for both low-level (e.g., increment, shift, addition) and high-level (e.g., square root, trigonometry, logarithm) arithmetic subroutines.

Assumption of Optimality: The work operates on the assumption that circuit designs minimising the T-count and qubit count in a baseline architecture will also yield the smallest active volume. This allows the author to focus on minimising T-counts (specifically Toffoli and temporary AND gates) rather than calculating active volumes for every possible design variation.
Gate Set Optimisation: The author utilises the "temporary AND" gate (introduced by Gidney) to halve the T-count of certain operations. They also opt to implement Toffoli and temporary AND gates using CCZ magic states rather than T states, as CCZ distillation is often cheaper in the active volume model.
Active Volume Calculation via Orientated ZX Diagrams:
- The author converts standard circuit diagrams into ZX diagrams and then into Orientated ZX (OZX) diagrams.
- OZX diagrams incorporate spatial awareness, mapping spiders to specific ports (U, D, N, S, E, W) of 3D spacetime blocks.
- By applying ZX calculus rules (such as spider merging) within the constraints of OZX connectivity, the author simplifies the diagrams to count the minimum number of logical blocks (spiders) required.
- The total active volume is calculated by summing the block counts of the spiders and adding the distillation costs for any required magic states (e.g., CCZ states).
Reaction Depth Analysis: The paper accounts for "reaction depth," the time required for classical processing of measurement outcomes to determine corrective operations for non-Clifford gates. The total reaction depth is calculated as the sum of the depths of sequential components, providing an upper bound on the computation time.

Key Contributions and Results

1. Low-Level Arithmetic Subroutines
The paper presents optimised circuits for fundamental operations, verified via Qiskit simulation:

Increment & Controlled Increment: Modified to use temporary AND gates, reducing the T-count to $7n - 14$ (for increment) and $7n - 7$ (for controlled increment).
Shifts: Demonstrated that in active volume architectures, uncontrolled shifts are "free" (zero logical block cost) via relabeling. Controlled shifts are optimised using ZX spider mergers, though the reduction in block count is marginal compared to the non-optimised version.
Controlled Addition/Subtraction (CAS): A new Gidney-style controlled adder is designed. It achieves an $\approx 8\%$ reduction in active volume compared to a straightforward summation of costs, with a T-count of $7n - 7$ .
Multiplication: A new multiplier using a Toffoli array and Controlled Overflow Gidney (COG) adders is proposed. It achieves a 62% reduction in the leading factor of the T-count compared to the state-of-the-art design by Muñoz-Coreas & Thapliyal [21].
Squaring: Optimised by trading qubit count for CNOT gates, reducing the qubit requirement from $6n+2$ to $5n$ for multiplication, with specific adjustments for squaring.

2. High-Level Arithmetic Subroutines
The author extends these low-level components to complex functions:

Square Root: An optimised non-restoring square root algorithm is presented. It reduces the T-count significantly compared to the leading design [19], though it requires slightly more qubits ( $3n$ vs $2n+1$ ).
Trigonometry (Piecewise Polynomial Evaluation - PPE): The paper adapts Häner's PPE circuit. By replacing the original adders and multipliers with the author's optimised versions (OG adder, COG multiplier) and a new comparator, the T-count is reduced by approximately 2/3 in the leading terms compared to the original implementation [20].
Inverse Trigonometry (Arcsine): A circuit combining polynomial evaluation and square root is constructed. The new design achieves an almost 6-fold reduction in the $n^2$ term of the T-count compared to Häner's implementation [20].
Logarithm: An implementation of Wang et al.'s qFBE method is adapted for active volume computing. The paper provides the first active volume and T-count estimates for this method, noting it is $O(n^3)$ in quantum operations.

3. Structural Sensitivity of Active Volume
A critical finding is that circuit structure impacts active volume independently of gate counts. The author demonstrates that logically equivalent circuits with identical gate counts can have different active volumes due to the arrangement of controls and targets. Specifically, designs that maximise uninterrupted lines of controls and targets allow for more efficient spider mergers in the ZX calculus, reducing the block count. This suggests that optimising for gate count alone is insufficient for active volume architectures.

Significance and Claims

The paper claims to provide a "simple methodology" for estimating and optimising active volumes using orientated ZX diagrams. Its primary significance lies in:

Resource Reduction: Significantly reducing T-counts and active volumes for a wide range of arithmetic subroutines, which are the building blocks for major quantum algorithms (e.g., factoring, HHL).
Architectural Awareness: Highlighting that the "active volume" metric is sensitive to circuit topology, not just gate complexity. This necessitates a shift in how circuits are designed for architectures with long-range connectivity.
Catalogue Expansion: Providing a necessary resource of optimised subroutines (including high-level functions like $\log x$ and $\arcsin x$ ) that were previously lacking in the context of active volume computing.

The author concludes that while their work focuses on T-counts and active volumes, future work should also consider Clifford gate optimisation and the inclusion of other low-level subroutines like carry look-ahead adders and division circuits. They do not claim to have solved all resource estimation problems but aim to prompt further investigation into AV-aware circuit design.

Problem Statement

Methodology

Key Contributions and Results

Significance and Claims

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