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# Model quantum computer and quantum code:

In this section we describe a simple model for a quantum computer based on a classical computer instructing a machine to manipulate a set of spins. This model has some intrinsic limitations which make designing algorithms in a high-level language somewhat tricky. We discuss some of the rules for writing such quantum computer code as a high-level language and give an example.

Consider the following model for the operation of a quantum computer: Several thousand spin- particles (or two-level systems) are initially in some well defined state, such as spin-down. A classical machine takes single spins or pairs of spins and entangles them (performing an elementary one-bit operation or the two-bit XOR gate); see Fig. 9a, b and c. These stages are repeated on different pairs of spins according to the instructions of a conventional computer program. Since the spins are entangled, we must not look at the spins at intermediate stages: We must keep the quantum superposition intact. Furthermore, nothing else may interfere with the spins which could destroy their orientation or interrupt their unitary evolution. Once this well-defined cycle of manipulation is complete the orientations of the spins are measured (Fig. 9d). This set of measured orientations is the output of the computation. Fig. 9 Model quantum computer as pictured by Shor . Initially all particles are spin-down. Stage a) a classical machine takes a single or pair of spins and in stage b) it performs a selected one-bit or two-bit operation; in stage c) the ``entangled'' particles are returned to their original locations. These three stages are repeated many times in accord with the instructions given by an ordinary classical computer. When this cycle is complete stage d) consists of measuring the state of the particles (leaving them in some particular bit-string); this bit-string is the result of the computation.

Given this paradigm for a quantum computer, what might its high-level language (its computer code) look like? The most serious difficulty that must be dealt with is that the quantum information is manipulated by a conventional computer in a completely blind manner---without any access to the values of this quantum information. This means that the program cannot utilize ``shortcuts'' conditional on the value of a quantum variable (or register or bit). For example, loops must be iterated through exactly the same number of times independent of the values of the quantum variables. Similarly, conditional branches around large pieces of code must be broken down into repeated conditions for each step. In addition, each instruction performed upon the quantum bits must be logically reversible. Thus, ordinary assignments of a value to a variable, such as
| a = n, are not legal and must instead be performed as increments on an initially zeroed variable, such as | a = | a + n.

An example of such code that could run on this machine might look like this :

```       do 10 k = 1, worstdiv
| a = | a - n
if ( | a >= 0 ) | q = | q + 1
10 continue
do 20 k = 1, worstdiv
if ( k > | q ) | a = | a + n
20 continue

```
This code fragment could be used to calculate the quotient and the remainder, placed in and , respectively, for the division of by n; the constant worstdiv is the worst-case number of times the loop must be traversed. Here is initially zero. Each instruction here is either a conventional computer instruction or one involving some quantum variables. The former are direct instructions for the external computer, while the latter must be interpreted as a sequence of manipulations to be performed upon the quantum bits. As it stands, this code is not reversible (neither is it very efficient), e.g., the label 10 gives no specification of which routes might be used to get to it. It can, however, be easily rewritten .   Next: Quantum parallelism: Period Up: Quantum computation: a tutorial Previous: Logic gates for

Samuel L.~Braunstein
Wed Aug 23 11:54:31 IDT 1995