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arXiv:2002.10592 · IEEE International Symposium on Multiple-Valued Logic (ISMVL)

Efficient Quantum Circuit Decompositions via Intermediate Qudits

Jonathan M. Baker, Casey Duckering, Frederic T. Chong

Published Nov 1, 2020• 19 citations
arXivHTMLPDF

Abstract

Many quantum algorithms make use of ancilla, additional qubits used to store temporary information during computation, to reduce the total execution time. Quantum computers will be resource-constrained for years to come so reducing ancilla requirements is crucial. In this work, we give a method to generate ancilia out of idle qubits by placing some in higher-value states, called qudits. We show how to take a circuit with many O(n) ancilla and design an ancilla-free circuit with the same asymptotic depth. Using this, we give a circuit construction for an in-place adder and a constant adder both with O(log n) depth using temporary qudits and no ancilla.

DOI
10.1109/ismvl49045.2020.9345604

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Many quantum algorithms make use of ancilla, additional qubits used to store temporary information during computation, to reduce the total execution time. Quantum computers will be resource-constrained for years to come so reducing ancilla requirements is crucial. In this work, we give a method to generate ancilia out of idle qubits by placing some in higher-value states, called qudits. We show how to take a circuit with many O(n) ancilla and design an ancilla-free circuit with the same asymptotic depth. Using this, we give a circuit construction for an in-place adder and a constant adder both with O(log n) depth using temporary qudits and no ancilla.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qu trits . Past work with qutrits has demonstrated only constant factor improvements, owing to the log 2 (3) binary-to-ternary compression factor. We present a novel technique, intermediate qutrits, to achieve sublinear depth decompositions of the Generalized Toffoli and other arithmetic circuits using no additional ancilla—a significant improvement over linear depth for the best qubit-only equivalents. For example, our Generalized Toffoli construction features a 70× improvement in two-qudit gate count over a qubit-only decomposition. This results in circuit cost reductions for important algorithms like quantum neurons, Grover search, and even Shor’s algorithm. Using a previously developed simulator with near-term noise models, we demonstrate for these models over 90% mean reliability (fidelity) for the Toffoli construction, versus under 30% for the qubit-only baseline. For our other constructions, such as the Incrementer, the A + B adder and the +K adder, we demonstrate the power of intermediate qutrits in producing asymptotic depth improvements with no additional ancilla. Together, these results suggest qutrits offer a promising path toward scaling quantum computation.
We advocate for a fundamentally different way to perform quantum computation by using three-level qutrits instead of qubits. In particular, we substantially reduce the resource requirements of quantum computations by exploiting a third state for temporary variables (ancilla) in quantum circuits. Past work with qutrits has demonstrated only constant factor improvements, owing to the log2(3) binary-to-ternary compression factor. We present a novel technique using qutrits to achieve a logarithmic runtime decomposition of the Generalized Toffoli gate using no ancilla---an exponential improvement over the best qubit-only equivalent. Our approach features a 70x improvement in total two-qudit gate count over the qubit-only decomposition. This results in improvements for important algorithms for arithmetic and QRAM. Simulation results under realistic noise models indicate over 90% mean reliability (fidelity) for our circuit, versus under 30% for the qubit-only baseline. These results suggest that qutrits offer a promising path toward extending the frontier of quantum computers.
We construct an exact nnn-bit quantum adder with O(n)O(n)O(n) gates and O(log⁡n)O(\log n)O(logn) depth that requires no ancillary qubits. The circuit uses only XXX, CNOT, and Toffoli gates under all-to-all connectivity and is classically reversible. Our construction accelerates the controlled ladder underlying addition. Grouping the data into blocks and use existing control qubits as conditional workspace gives a recursive construction with O(nlog⁡log⁡n)O(n\log\log n)O(nloglogn) gates and O(log⁡n)O(\log n)O(logn) depth. We then coordinate inter-block propagation via two short-path forest operations to cut the per-block workspace requirement, yielding a nonrecursive implementation with O(n)O(n)O(n) gates. Both the gate count and circuit depth are asymptotically optimal.
Uncomputation of ancilla qubits is essential in quantum computing to reset auxiliary qubits to a zero state, ensuring their safe discarding and reducing resource overhead. Current methods for uncomputation either rely on manual procedures or use additional ancilla qubits to store intermediate computation results, which increases qubit storage requirements. Additionally, most in-place uncomputation techniques employ an all-or-nothing strategy, wherein they perform uncomputation only when all ancilla qubits in the given circuit can be uncomputed.In this work, we tackle the problem of uncomputing a maximum number of ancilla qubits in a quantum circuit with minimal qubit storage overhead. We propose a novel approach where partial uncomputation—reverting the last ’m’ gates—enables the uncomputation of more qubits. We present three algorithms designed to uncompute the largest possible set of ancilla qubits and demonstrate their effectiveness through experiments on various randomly generated quantum circuits.
We present quantum circuits for comparison and increment operations that achieve an asymptotically optimal gate count of Θ(n)Θ(n)Θ(n) and depth of Θ(log⁡n)Θ(\log n)Θ(logn) over the Clifford+Toffoli gate set, while using a provably minimal number of qubits. We extend these results to classical-quantum comparators, yielding an improved classical-quantum adder with an optimal qubit count. Given the ubiquity of these operations as algorithmic building blocks, our constructions translate directly into reduced circuit complexity for many quantum algorithms. As a notable example, they can be used to improve a space-efficient circuit for Shor's factoring algorithm, reducing circuit depth from O(n3)\mathcal{O}(n^3)O(n3) to O(n2log⁡2n)\mathcal{O}(n^2 \log^2 n)O(n2log2n) without increasing either the qubit count or the asymptotic gate complexity. Underpinning these results is a general theorem demonstrating how to trade ancilla qubits for control qubits with low overhead in both depth and gate count, providing a broadly applicable tool for quantum circuit design.
We improve the number of T gates needed to perform an n-bit adder from 8 n + O ( 1 ) to 4 n + 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 8 n + 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.