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arXiv:1512.03824

Improved Quantum Ternary Arithmetics

Alex Bocharov, Shawn Xingshan Cui, Martin Roetteler, Krysta M. Svore

Published Dec 11, 2015• 9 citations
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Abstract

Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more complicated form of the ternary carry, which leads to higher resource counts for implementations over a universal ternary gate set. Our ternary ripple adder circuit has a circuit depth of $O(n)$ and uses only $1$ ancilla, making it more efficient in both, circuit depth and width than previous constructions. Our ternary carry lookahead circuit has a circuit depth of only $O(\log\,n)$, while using with $O(n)$ ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate $C(X): |i,j\rangle \mapsto |i, j + δ_{i,2} \mod 3\rangle$ or from the two-qutrit Clifford gates and the non-Clifford gate $P_9=\mbox{diag}(e^{-2 π\, i/9},1,e^{2 π\, i/9})$. The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomial over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.

DOI
10.48550/arxiv.1512.03824

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Qutrit (or ternary) structures arise naturally in many quantum systems, particularly in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders, namely ripple carry adders and carry look-ahead adders. The main difference to the binary case is the more complicated form of the ternary carry, which leads to higher resource counts for implementations over a universal ternary gate set. Our ternary ripple adder circuit has a circuit depth of O(n)O(n)O(n) and uses only 111 ancilla, making it more efficient in both, circuit depth and width than previous constructions. Our ternary carry lookahead circuit has a circuit depth of only O(log⁡ n)O(\log\,n)O(logn), while using with O(n)O(n)O(n) ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate C(X):∣i,j⟩↦∣i,j+δi,2mod  3⟩C(X): |i,j\rangle \mapsto |i, j + δ_{i,2} \mod 3\rangleC(X):∣i,j⟩↦∣i,j+δi,2​mod3⟩ or from the two-qutrit Clifford gates and the non-Clifford gate P9=\mboxdiag(e−2π i/9,1,e2π i/9)P_9=\mbox{diag}(e^{-2 π\, i/9},1,e^{2 π\, i/9})P9​=\mboxdiag(e−2πi/9,1,e2πi/9). The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomial over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.
Qutrit (or ternary) structures arise naturally in many quantum systems, notably in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders. The main distinction from the binary adders is a richer ternary carry which leads potentially to higher resource counts in universal ternary bases. Our ternary ripple adder circuit has a circuit depth of O(n) and uses only 1 ancilla, making it more efficient in both, circuit depth and width, when compared with previous constructions. Our ternary carry lookahead circuit has a circuit depth of only O(log n), while using O(n) ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate C(X) : |i, ji 7→ |i, j + δi,2 mod 3i or from the two-qutrit Clifford gates and the non-Clifford gate P9 = diag(e −2π i/9 , 1, e 2π i/9 ). The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomials over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.
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.
Quantum computation relies on exploiting quantum mechanical phenomena, and has received significant attention in recent years. Higher-dimensional quantum systems increase the density of encoded information per computing element (e.g., qutrit for three-level system), resulting in less resource overhead. For instance, 63% reduction in the number of qutrits is possible for ternary quantum systems as compared to the corresponding binary systems. The proposed work exploits this fact to synthesize ternary reversible circuits employing a cycle-based technique. The method starts from the ternary reversible specification of a given function in the form of a permutation. The permutation cycles are factored into simpler three-cycles and two-cycles, which are then mapped to ternary reversible gates. Different gate libraries are used to synthesize three-cycles and two-cycles, respectively. A gate decomposition approach is also proposed to synthesize a quantum gate netlist in terms of elementary ternary quantum gates, viz. Muthukrishnan–Stroud gate and shift gate. Synthesis results on benchmark functions indicate that the proposed method results in 27% and 6% improvements in quantum cost and gate count, respectively, over existing works in the literature.
The state of the art of quantum circuits using the ripple-carry strategy for the addition and comparison of two n-bit numbers is presented, as well as optimizations in the Clifford+ <?TeX T\mathsf {T}T?> Math 1 gate set, both in terms of <?TeX CNOT\mathsf {CNOT}CNOT?> Math 2 -depth and <?TeX T\mathsf {T}T?> Math 3 -depth, or <?TeX CNOT\mathsf {CNOT}CNOT?> Math 4 -count and <?TeX T\mathsf {T}T?> Math 5 -count. In particular, we consider the adders presented by Cuccaro et al. and Takahashi et al., and exhibit an adder with a <?TeX T\mathsf {T}T?> Math 6 -depth of 3n and a <?TeX CNOT\mathsf {CNOT}CNOT?> Math 7 -depth of 8n, while without optimization of the original circuits, a <?TeX T\mathsf {T}T?> Math 8 -depth of 6n is expected. Note that we have focused here on quantum ripple-carry adders using at most one ancilla, without any approximation of the 3-qubit gates involved (Toffoli, Peres and TR) or any strategy involving a measurement.
Efficient Synthesis of Multi-Controlled Toffoli Gates with Ternary Clifford+P9+P_9+P9​ Gates
Temporary occupation of qutrit levels can reduce the width and depth required for binary circuit logic in quantum computing. We introduce an efficient intermediate-qutrit decomposition of multi-controlled Toffoli gates with binary-subspace inputs and outputs. For balanced control widths n=2h−1n=2^h-1n=2h−1, the proposed decomposition is evaluated hierarchically through a tree, allowing independent subtree computations to proceed in parallel and reducing the depth from linear to logarithmic. The balanced low-depth construction requires 6n+36n+36n+3 logical P9P_9P9​ injections and n−34\frac{n-3}{4}4n−3​ clean ancillary qutrits. It matches the direct P9P_9P9​ count of a recursively extended Clifford+P9P_9P9​ baseline derived from the state-of-the-art work while using asymptotically one-quarter as many clean ancillas and replacing linear depth with logarithmic depth. For arbitrary control widths, a sequential extension preserves the 6n+36n+36n+3 P9P_9P9​ count but has depth O(log⁡m+n−m)O(\log m+n-m)O(logm+n−m) for a balanced core m=2h−1≤nm=2^h-1\leq nm=2h−1≤n, which is linear in the worst case over nnn. By reducing the resource overhead of a widely used reversible primitive, the proposed decomposition provides a practical building block for the design and compilation of more resource-efficient quantum algorithms in the fault-tolerant regime.
Reversible logic has represented itself as a prominent technology which plays an important role in Quantum Computing. Theoretically Quantum Computers operates at high speed and consumes less power. Furthermore, Reversible logic can break the conventional speed of power trade-off. To prove this we are implementing Ripple Carry Adder and Carry Look Ahead Adder using reversible logic gates. The paper presents efficient adder circuits using Peres gate, New fault Tolerant gate and Double Feyman gate. The complexity, simulated outputs and the speed parameters for the adder circuit have been indicated using the Quartus II 9.1 edition and Modelsim tool. Moreover, the quantum algorithms can potentially solve NP-complete problems. Furthermore, doing high performance functions beyond the limit of deterministic computer systems is possible by only reversible logic. Quantum operations are unitary in nature which is reversible and hence the arithmetic operations like adders can be implemented using the reversible logic. Likewise, optimized block size, minimum ancillary bits, and one to one mappings of inputs and outputs are obtained in the reversible adder circuit, giving fan out as 1. Hence, the Ripple Carry Adder and Carry look Ahead Adder is providing complexity reduction and high speed, proving to be an efficient and important part in the development of arithmetic blocks of Quantum Computers.