Quantum Random Walks, which have drawn much attention over the past few decades for their distinctly non-classical behavior, is a promising subfield within Quantum Computing. Theoretical framework and applications for these walks have seen many great mathematical advances, with experimental demonstrations now catching up. In this study, we examine the viability of implementing Coin Quantum Random Walks using a Quantum Adder based Shift Operator, with quantum circuit designs specifically for superconducting qubits. We focus on the strengths and weaknesses of these walks, particularly circuit depth, gate count, connectivity requirements, and scalability. We propose and analyze a novel approach to implementing boundary conditions for these walks, demonstrating the technique explicitly in one and two dimensions. And finally, we present several fidelity results from running our circuits on IBM's quantum volume 32 `Toronto' chip, showcasing the extent to which these NISQ devices can currently handle quantum walks.
Control modular addition is a core arithmetic function, and we must consider the computational cost for actual quantum computers to realize efficient implementation. To achieve a low computational cost in a control modular adder, we focus on minimizing KQ, defined by the product of the number of qubits and the depth of the circuit. In this paper, we construct an efficient control modular adder with small KQ by using relative-phase Toffoli gates in two major types of quantum computers: Fault-Tolerant Quantum Computers (FTQ) on the Logical layer and Noisy Intermediate-Scale Quantum Computers (NI…
The quantum adder is an essential attribute of a quantum computer, just as classical adder is needed for operation of a digital computer. We model the quantum full adder as a realistic complex algorithm on a large number of qubits in an Ising-spin quantum computer. Our results are an important step toward effective modeling of the quantum modular adder which is needed for Shor's and other quantum algorithms. Our full adder has the following features: (i) The near-resonant transitions with small detunings are completely suppressed, which allows us to decrease errors by several orders of magnitu…
Some important VLSI adder circuits are implemented using quantum dots (qd) and Spin Polarized Scanning Tunneling Microscopy (SPSTM) in Single Spin Logic (SSL) paradigm. A simple comparison between these adder circuits shows that the mirror adder implementation in SSL does not carry any advantage over CMOS adder in terms of complexity and number of qds, opposite to the trend observed in their charge-based counterparts. On the contrary, the transmission gate adder, Static and Dynamic Manchester carry gate adders in SSL reduce the complexity and number of qds, in harmony with the trend shown in t…
Quantum computers require quantum processors. An important part of the processor of any computer is the arithmetic unit, which performs binary addition, subtraction, division and multiplication, however multiplication can be performed using repeated addition, while division can be performed using repeated subtraction. In this paper we present two designs using the reversible $R^3$ gate to perform the quantum half adder/ subtractor and the quantum full adder/subtractor. The proposed half adder/subtractor design can be used to perform different logical operations, such as $AND$, $XOR$, $NAND$, $…
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, …
In this paper, two quantum networks for the addition operation are presented. One is the Modified Quantum Plain (MQP) adder, and the other is the Quantum Carry Look-Ahead (QCLA) adder. The MQP adder is obtained by modifying the Conventional Quantum Plain (CQP) adder. The QCLA adder is an extension of conventional digital Carry Look-Ahead adder. Compared with the CQP adder, two main advantages are as follows: First, the proposed MQP and QCLA adders have less number of elementary gates than the CQP adder. Secondly, the number of processing stages of the MQP and QCLA adder are less than ones of t…
Quantum addition circuits are considered being of two types: 1) Toffolli-adder circuits which use only classical reversible gates (CNOT and Toffoli), and 2) QFT-adder circuits based on the quantum Fourier transformation. We present the first systematic translation of the QFT-addition circuit into a Toffoli-based adder. This result shows that QFT-addition has fundamentally the same fault-tolerance cost (e.g. T-count) as the most cost-efficient Toffoli-adder: instead of using approximate decompositions of the gates from the QFT circuit, it is more efficient to merge gates. In order to achieve th…
Quantum circuits of arithmetic operations such as addition are needed to implement quantum algorithms in hardware. Quantum circuits based on Clifford+T gates are used as they can be made tolerant to noise. The tradeoff of gaining fault tolerance from using Clifford+T gates and error correcting codes is the high implementation overhead of the T gate. As a result, the T-count performance measure has become important in quantum circuit design. Due to noise, the risk for errors in a quantum circuit computation increases as the number of gate layers (or depth) in the circuit increases. As a result,…
Progress in quantum hardware design is progressing toward machines of sufficient size to begin realizing quantum algorithms in disciplines such as encryption and physics. Quantum circuits for addition are crucial to realize many quantum algorithms on these machines. Ideally, quantum circuits based on fault-tolerant gates and error-correcting codes should be used as they tolerant environmental noise. However, current machines called Noisy Intermediate Scale Quantum (NISQ) machines cannot support the overhead associated with faulttolerant design. In response, low depth circuits such as quantum c…
We present a new linear-depth ripple-carry quantum addition circuit. Previous addition circuits required linearly many ancillary qubits; our new adder uses only a single ancillary qubit. Also, our circuit has lower depth and fewer gates than previous ripple-carry adders.
In this work, we present a class of new designs for reversible binary and BCD adder circuits. The proposed designs are primarily optimized for the number of ancilla inputs and the number of garbage outputs and are designed for possible best values for the quantum cost and delay. First, we propose two new designs for the reversible ripple carry adder: (i) one with no input carry $c_0$ and no ancilla input bits, and (ii) one with input carry $c_0$ and no ancilla input bits. The proposed reversible ripple carry adder designs with no ancilla input bits have less quantum cost and logic depth (delay…
We give a new ``nested adds'' circuit for implementing Shor's algorithm in linear width and quadratic depth on a nearest-neighbor machine. Our circuit combines Draper's transform adder with approximation ideas of Zalka. The transform adder requires small controlled rotations. We also give another version, with slightly larger depth, using only reversible classical gates. We do not know which version will ultimately be cheaper to implement.
We study the application of one nanoring driven by a laser field in different states of polarization in logic circuits. In particular we show that assigning boolean values to different state of the incident laser field and to the emitted signals, we can create logic gates such as OR, XOR and AND. We also show the possibility to make logic circuits such as half-adder and full-adder using one and two nanoring respectively. Using two nanorings we made the Toffoli gate. Finally we use the final angular momentum acquired by the electron to store information and hence show the possibility to use an …
Reciprocal Quantum Logic combines the speed and power-efficiency of single-flux quantum superconductor devices with design features that are similar to CMOS. We have demonstrated an 8-bit carry look-ahead adder in the technology using combinational gates with fanout of four and non-local interconnect. Measured power dissipation of the fully active circuit is only 510 nW at 6.2GHz. Latency is only 150 ps at a clock rate of 10GHz.
In this work, we propose an adder for the 2D NTC architecture, designed to match the architectural constraints of many quantum computing technologies. The chosen architecture allows the layout of logical qubits in two dimensions and the concurrent execution of one- and two-qubit gates with nearest-neighbor interaction only. The proposed adder works in three phases. In the first phase, the first column generates the summation output and the other columns do the carry-lookahead operations. In the second phase, these intermediate values are propagated from column to column, preparing for computat…
Quantum arithmetic circuits have practical applications in various quantum algorithms. In this paper, we address quantum addition on 2-dimensional nearest-neighbor architectures based on the work presented by Choi and Van Meter (JETC 2012). To this end, we propose new circuit structures for some basic blocks in the adder, and reduce communication overhead by adding concurrency to consecutive blocks and also by parallel execution of expensive Toffoli gates. The proposed optimizations reduce total depth from $140\sqrt n+k_1$ to $92\sqrt n+k_2$ for constants $k_1,k_2$ and affect the computation f…
We present an arithmetic circuit performing constant modular addition having $\mathcal{O}(n)$ depth of Toffoli gates and using a total of $n+3$ qubits. This is an improvement by a factor of two compared to the width of the state-of-the-art Toffoli-based constant modular adder. The advantage of our adder, compared to the ones operating in the Fourier-basis, is that it does not require small angle rotations and their Clifford+T decomposition. Our circuit uses a recursive adder combined with the modular addition scheme proposed by Vedral et. al. The circuit is implemented and verified exhaustivel…
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 use…
It is well-known that any quantum gate can be decomposed into the universal gate set {T, H, CNOT}, and recent results have shown that each of these gates can be implemented using a dynamic quantum walk, which is a continuous-time quantum walk on a sequence of graphs. This procedure for converting a quantum gate into a dynamic quantum walk, however, can result in long sequences of graphs. To alleviate this, in this paper, we develop a length-3 dynamic quantum walk that implements any single-qubit gate. Furthermore, we extend this result to give length-3 dynamic quantum walks that implement any …
Quantum circuits of many qubits are extremely difficult to realize; thus, the number of qubits is an important metric in a quantum circuit design. Further, scalable and reliable quantum circuits are based on Clifford + T gates. An efficient quantum circuit saves quantum hardware resources by reducing the number of T gates without substantially increasing the number of qubits. Recently, the design of a quantum multiplier is presented by Babu [1] which improves the existing works in terms of number of quantum gates, number of qubits, and delay. However, the recent design is not based on fault-to…
We investigate the counterparts of random walk in universal quantum computing and their implementation using standard quantum circuits. Quantum walk have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1-D graph, namely a circle, and show how to implement discrete circular quantum walk in quantum circuits built with universal CNOT and single quit gates. We review elementary quantum gates and circuit decomposition and propose a a generalized version of the all CNOT based quantum discrete circular walk. We simulated these circuits on…
It is demonstrated that in gate-based quantum computing architectures quantum walk is a natural mathematical description of quantum gates. It originates from field-matter interaction driving the system, but is not attached to specific qubit designs and can be formulated for very general field-matter interactions. It is shown that, most generally, gates are described by a set of coined quantum walks. Rotating wave and resonant approximations for field-matter interaction simplify the walks, factorizing the coin, and leading to pure continuous time quantum walk description. The walks reside on a …
The physics of quantum walks on graphs is formulated in Hamiltonian language, both for simple quantum walks and for composite walks, where extra discrete degrees of freedom live at each node of the graph. It is shown how to map between quantum walk Hamiltonians and Hamiltonians for qubit systems and quantum circuits; this is done for both a single- and multi-excitation coding, and for more general mappings. Specific examples of spin chains, as well as static and dynamic systems of qubits, are mapped to quantum walks, and walks on hyperlattices and hypercubes are mapped to various gate systems.…
A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n-arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is de…
It was recently shown that continuous-time quantum walks on dynamic graphs, i.e., sequences of static graphs whose edges change at specific times, can implement a universal set of quantum gates. This result treated all isolated vertices as having self-loops, so they all evolved by a phase under the quantum walk. In this paper, we permit isolated vertices to be loopless or looped, and loopless isolated vertices do not evolve at all under the quantum walk. Using this distinction, we construct simpler dynamic graphs that implement the Pauli gates and a set of universal quantum gates consisting of…
Non-local higher-energy auxiliary states have been successfully used to entangle pairs of qubits in different quantum computing systems. Typically a longer-span non-local state or sequential application of few-qubit entangling gates are needed to produce a non-trivial multiqubit gate. In many cases a single non-local state that span over the entire system is difficult to use due to spectral crowding or impossible to have. At the same time, many multiqubit systems can naturally develop a network of multiple non-local higher-energy states that span over few qubits each. We show that continuous t…
In this work, we show that ma-QAOA is equivalent to a restriction of continuous-time quantum walks on dynamic graphs. We then show it is universal for computation by finding the appropriate $B$ and $C$ operators and angles that implement the universal gate set consisting of the Hadamard, $π/8$ and Controlled-Not gates in the ma-QAOA framework. This result begins to bridge the gap between the continuous-time quantum walk model and gate model of quantum computation.
Quantum Hamiltonian Computing is a recent approach that uses quantum systems, in particular a single molecule, to perform computational tasks. Within this approach, we present explicit methods to construct logic gates using two different designs, where the logical outputs are encoded either at fixed energy and spatial positioning of the quantum states, or at different energies. We use these results to construct quantum Boolean adders involving a minimal number of quantum states with the two designs. We also establish a matrix algebra giving an analogy between classical Boolean logic gates and …
Multiplier circuits play an important role in reversible computation, which is helpful in diverse areas such as low power CMOS design, optical computing, DNA computing and bioinformatics. Here we propose a new reversible multiplier circuit with optimized hardware complexity. The optimized multiplier circuit is compared with the earlier proposals. We have shown that the quantum cost of earlier proposals can be further reduced with the help of existing local optimization algorithms (e.g. template matching, moving rule and deletion rule). A systematic protocol for reduction of quantum cost has be…
We describe an array of quantum gates implementing Shor's algorithm for prime factorization in a quantum computer. The array includes a circuit for modular exponentiation with several subcomponents (such as controlled multipliers, adders, etc) which are described in terms of elementary Toffoli gates. We present a simple analysis of the impact of losses and decoherence on the performance of this quantum factoring circuit. For that purpose, we simulate a quantum computer which is running the program to factor N = 15 while interacting with a dissipative environment. As a consequence of this inter…
Synthesis of quaternary quantum circuits involves basic quaternary gates and logic operations in the quaternary quantum domain. In this paper, we propose new projection operations and quaternary logic gates for synthesizing quaternary logic functions. We also demonstrate the realization of the proposed gates using basic quantum quaternary operations. We then employ our synthesis method to design of quaternary adder and some benchmark circuits. Our results in terms of circuit cost, are better than the existing works.
Quantum walk has been regarded as a primitive to universal quantum computation. By using the operations required to describe the single particle discrete-time quantum walk on a position space we demonstrate the realization of the universal set of quantum gates on two- and three-qubit systems. The idea is to utilize the effective Hilbert space of the single qubit and the position space on which it evolves in order to realize multi-qubit states and universal set of quantum gates on them. Realization of many non-trivial gates and engineering arbitrary states is simpler in the proposed quantum wal…
