Halving the cost of quantum addition (1709.06648v3)

Abstract: We improve the number of T gates needed to perform an n-bit adder from 8n + O(1) to 4n + O(1). We do so via a "temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of 8n + O(1), a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.

• The paper presents a method to halve the T-count for n-bit quantum adders, reducing resource cost from $8n+O(1)$ to $4n+O(1)$.
• This T-count reduction is achieved using a novel temporary logical-AND operation that requires only four T gates to compute and none to uncompute.
• The reduced T-count enables more efficient quantum algorithms and improves feasibility for surface code architectures where T-gates are expensive.

An Analysis of "Halving the cost of quantum addition"

The field of quantum computation is increasingly focused on optimizing resource usage in quantum algorithms, particularly concerning quantum gates. In the paper titled "Halving the cost of quantum addition" by Craig Gidney, the author presents a significant advancement in reducing the T-count, a critical factor in the resource cost of quantum computations, specifically for the problem of quantum addition.

At the core of this paper is the reduction of the T-count required for an $n$-bit adder from $8n + O(1)$ to $4n + O(1)$. This improvement is achieved through the innovative use of temporary logical-AND operations, which enable logical AND operations between qubits with reduced T-gate usage. Notably, the process to compute this logical-AND uses only four T gates, while its subsequent uncomputation is achieved without any T gate expense. The application of this method is portrayed as more versatile than previously realized in quantum computations, with potential utility across several areas such as integer and modular arithmetic, rotation synthesis, the quantum Fourier transform, and algorithms like Shor’s algorithm and Grover’s search.

The resource-efficient construction proposed by Gidney has profound implications for implementations based on the surface code, a popular quantum error correction scheme suitable for 2D qubit arrays. As T gates, which are typically costly due to the need for state distillation, dominate the computational resources needed in surface code-based architectures, reducing their count significantly impacts the projected cost and feasibility of quantum computations.

Specifically, the paper introduces an optimized $n$-bit controlled adder circuit, which maintains a T-count of $8n + O(1)$, while the out-of-place adder offered in the work can be uncomputed without T gates. Such developments open pathways for more efficient designs of higher-level quantum algorithms that rely heavily on arithmetic operations, thereby facilitating faster and less resource-intensive computations.

One of the noteworthy discussions in the paper is the opportunity cost associated with the ancillae used in these circuits. The trade-off between using ancilla qubits and the benefit of reducing T-count is crucial to the optimization of quantum circuit design. By addressing the 'effective' T-count, the paper provides insights into when and how employing additional ancillae can be advantageous, particularly on systems where the availability of T factories constrains the computation speed more significantly than qubit count.

Furthermore, the proposed methods introduce a strategy to significantly streamline operations involving large-scale number computations, providing a reduced T-count technique for operations like $R_Z(\theta)$ rotations and phase gradients. These operations are fundamental to the function of many quantum algorithms, widening the impact of the new construction beyond basic arithmetic.

In addition to addressing practical implementations, Gidney speculates on future research directions, acknowledging the theoretical bounds on the minimal T-count for quantum circuits implementing Toffoli gates. This work, therefore, stimulates further exploration into optimizing quantum gate implementations, promising substantial improvements not only at the gate level but also at more complex quantum circuit levels.

In conclusion, the advancements detailed in this paper hold significant potential for both practical and theoretical developments in quantum computing. By halving the T-count for quantum addition, and highlighting a broadly applicable logical-AND operation, Craig Gidney's work presents a substantive contribution to quantum algorithm efficiency. This research not only enhances current methodologies but also sets a foundation for continued evolution in optimizing quantum computational resources.