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CRC Press eBooks

Quantum Mechanical Computers

Richard P. Feynman

Published Jul 3, 2018• 1 citation
Publisher

Abstract

In this chapter, the authors discuss the part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. A more elaborate circuit is a full adder which takes a carry, c , and adds it to the two lines a and b and has an additional line, d, with a 0 input. Calculations can be done with reversible machines containing only reversible primitives. There are many sources of imperfections in the machine, but the first one people would like to consider is the possibility that the coefficients in the couplings, along the program line, are not exactly equal. The switches are represented by diagonal lines, and in boxes put the other matrices that operate on registers such as the NOT b .

DOI
10.1201/9780429500442-6

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In this chapter, the authors discuss the part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. A more elaborate circuit is a full adder which takes a carry, c , and adds it to the two lines a and b and has an additional line, d, with a 0 input. Calculations can be done with reversible machines containing only reversible primitives. There are many sources of imperfections in the machine, but the first one people would like to consider is the possibility that the coefficients in the couplings, along the program line, are not exactly equal. The switches are represented by diagonal lines, and in boxes put the other matrices that operate on registers such as the NOT b .
In this chapter, the authors discuss a part in an effort to analyze the physical limitations of computers due to the laws of physics. It is a result of computer science that a universal computer can be made by a suitably complex network of interconnected primitive elements. There are, however, a number of questions that the people would like to discuss in more detail such as the question of imperfections. It is analogous to the use of transistors in conventional machines where the people don't properly use all the analog continuum of the behavior of transistors, but just try to run them as saturated on or off digital devices so the logical analysis of the system behavior is easier. Of course the large system obeys quantum mechanics, but it is in interaction with the heat baths and other things that could make it effectively irreversible.
A computer is imagined in which all the parts are simple two-state objects obeying quantum mechanical laws. A time-independent Hamiltonian for such a system is written and analyzed. The computation is done in logically reversible steps. It is used to study possible limitations on computers due to fundamental physical laws. The ideal machine can run ballistically, coasting through the calculation in a short time TB. If there are imperfections due to irregularities in construction or finite temperatures the forward moving calculation is often reversed or scattered. External work must be done to drag the computer forward. The free energy lost in doing a calculation is kT In2 per logical step divided by turn factors. One is the mean free number of steps per scattering, measuring how perfectly the machine is built. The other is the mean time (divided by TB) that the calculation actually takes, which measures how gently you urge it forward.
In the preceding chapter we established that a suitable set of quantum gates, complemented by quantum error correction, allows us to produce a desired multiqubit unitary transformation. This transformation is one of the three steps in a quantum computation; the others, of course, are the preparation of the qubits in their initial state and the measurement of them after the transformation has been implemented. A quantum computation is designed to solve a problem or class of problems. The power of quantum computers is that they can do this, at least for some problems, very much more efficiently and quickly than any conventional computer based on classical logic operations. If we can build a quantum computer then a number of important problems which are currently intractable will become solvable. The potential for greatly enhanced computational power is, in itself, reason enough to study quantum computers, but there is another. Moore’s law is the observation that the number of transistors on a chip doubles roughly every eighteen months. A simple corollary is that computer performance also doubles on the same timescale. Associated with this exponential improvement is a dramatic reduction in the size of individual components. If the pace is to be kept up then it is inevitable that quantum effects will become increasingly important and ultimately will limit the operation of the computer. In these circumstances it is sensible to consider the possibility of harnessing quantum effects to realize quantum information processors and computers. We start with a brief introduction to the theory of computer science, the principles of which underlie the operation of what we shall refer to as classical computers. These include all existing machines and any based on the manipulation of classical bits. The development of computer science owes much to Turing, who devised a simple but powerful model of a computing device: the Turing machine. It its most elementary form, this consists of four elements. (i) A tape for data storage, which acts as a memory. This tape has a sequence of spaces, each of which has on it one of a finite set of symbols. (ii) A processor, which controls the operations of the machine.
Reversible ternary circuit design has attracted considerable attention because ternary logic can reduce interconnection ‎complexity relative to binary logic and is compatible with emerging quantum-computing and nanotechnology platforms. This ‎study proposes a reversible 2-trit ternary parallel adder with a quantum cost of 23, one constant input, and three garbage ‎outputs. Compared with previously reported counterparts, the proposed adder requires lower quantum cost and fewer ‎constant inputs and garbage outputs. Based on this adder, a reversible arithmetic unit is developed for two 2-trit unsigned ‎ternary numbers. The unit performs six arithmetic operations: A+B, A−B, A+1, A−1, A+B+1, and A−B−1. The unsigned inputs ‎and outputs are represented within the decimal range from 0 to 8. A reversible arithmetic unit for two signed 2-trit ternary ‎numbers is also proposed. This circuit has a quantum cost of 32 and requires only one constant input. It performs the same six ‎arithmetic operations using the 3’s-complement representation over the range from −4 to +4. In addition, a new reversible ‎overflow detection module is introduced for signed addition and subtraction. By integrating this module with the proposed ‎signed arithmetic unit, a complete reversible ternary arithmetic circuit with overflow-detection capability is obtained. All ‎proposed circuits are constructed using 1-qutrit shift gates and 2-qutrit Muthukrishnan–Stroud gates, which are primitive ‎ternary gates suitable for implementation in ion-trap quantum-computing technology.‎
Stimulated Raman adiabatic passage (STIRAP) is a well-studied pump-probe control scheme for manipulating the population of quantum states of atoms or molecules. By encoding the digits to be operated on as ``on'' or ``off'' laser input signals we show how STIRAP can be used to implement a finite-state logic machine. The physical conditions required for an effective STIRAP operation are related to the physical conditions expected for a logic machine. In particular, a condition is derived on the mean number of photons that represent an on pulse. A finite-state machine computes Boolean expressions that depend both on the input and on the present state of the machine. With two input signals we show how to implement a full adder where the carry-in digit is stored in the state of the machine. Furthermore, we show that it is possible to store the carry-out digit as the next state and thereby return the machine to a state ready for the next full addition. Such a machine operates as a cyclical full adder. We further show how this full adder can equally well be operated as a full subtractor. To the best of our knowledge this is the first example of a nanosized system that implements a full subtraction.
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.