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The Mathematics Behind Quantum Computing: Part IIThe second of two feature columns on this subject (the first is here)...
The Fast Fourier Transform is fast, but not fast enough for realistic factorization of large numbers. Factoring an n-bit number would require 3n·2n operations; that number increases exponentially with n. This month's column will examine how for a quantum computer this growth could be made polynomial, and the factorization problem could become tractable.
Data in a quantum computer are stored in qubits, and manipulated by gates.
Example: the NOT-gate. This is the simplest example of a gate that does something.
Here the operator is described by its values on the basis vectors. Note that NOT takes |0> + |1> to itself, so not every vector gets "negated." Also note that this operation is its own inverse, so it is automatically unitary.
Example: the R-gate. In terms of the basis vectors:
This gate has been called the "quantum coin-flip." If a qubit starts in state |0>, after the gate if it is measured it will return 0 with probability 1/2, and 1 with probability 1/2. This operation is also its own inverse.
Suppose the gate is acting on the two qubits qubit with state space basis |0>, |1> and qubit with state space basis |0>, |1>. The tensor product a b of a = a0|0> + a1|1> with b = b0|0> + b1|1> is a 4-component object best represented by the matrix:
In terms of the bases |0>, |1> and |0>, |1>, the tensor products
form a basis for the tensor product of the state spaces.
Example: Controlled-NOT Such a device inputs 2 qubits, say qubit and qubit; it leaves qubit unchanged and reverses the state of qubit whenever qubit is in state |1>.
This operation can also be described by saying that the sum modulo 2 of qubit and qubit is stored in qubit, and that qubit is carried along to make the operation reversible.
A typical element of the tensor product of the qubit state space (basis |0>, |1>) and the qubit state space (basis |0>, |1>) will have the form:
or, written in matrix notation,
subject to the condition |c0,0|2 + |c1,0|2 + |c0,1|2 + |c1,1|2 = 1.
Such an element will in general not be the tensor product ab of a = a0|0> + a1|1> and b = b0|0> + b1|1>. In fact, we can write a matrix as a tensor product
only when the determinant c0,0 c1,1 – c1,0 c0,1 is 0.
A state in the 2-qubit state space which is not of the form ab is called an entangled state. Here is why:
Parallel processing in a quantum computer
and we modify the Controlled-NOT gate so that if qubit starts in state |0>, then its output state is the modulo 2 sum of the input states of qubit and qubit; the other qubits remain unchanged. The outputs when qubit starts in state |1> will not concern us, except inasmuch as they make the whole operation unitary. Let us call this 3-input, 3-output gate a D-gate. In terms of the standard basis for the tensor product of the three state spaces, it is described by the table
To perform mod 2 addition in parallel, we start with the register in state |0> |0> |0>, and we apply the R-gate to qubit and to qubit. This leaves qubit in state (|0> + |1>) and qubit in state (|0> + |1>), and the register in state
We now apply the D-gate once to this register. The output is
Now each qubit, qubit state pair is entangled with its mod-2 sum. If we measure the first two qubits, and then measure the third, it will report the sum of the first two measurements.
The Quantum Fourier Transform
A quantum Fourier transform was first worked out by Peter Shor, in 1994. This refinement (which corresponds exactly to the the Radix-2 Cooley-Tukey algorithm) was discovered, almost immediately afterwards and independently, by Richard Cleve, Don Coppersmith and David Deutsch. Working with a register of q qubits
it uses a combination of the gates Rj (the R-gate applied to qubitj) with a set of reversible gates called Sj,k in Shor's notation. The gate Sj,k operates on the pair qubitj, qubitk, with j < k, as follows:
where ωk–j = e i π /2k–j. This number is a primitive (2k–j)-th root of –1.
For 3 bits, the QFT algorithm specifies the sequence
More specifically, suppose in our 3-qubit example that we carry along the input register as the first 3 qubits in the computation (this will turn out not be necessary). Then at the end if the first qubits are measured and report binary 5, i.e. |1>|0>|1>, the last three qubits must contain the superposition:
2–3/2 (|0>|0>|0> – |1>|0>|0> + i|0>|1>|0> – i|1>|1>|0> – e iπ/4 |0>|0>|1>
In the "row" corresponding to binary 5, the coefficient of the state "binary c" is exactly (2–3/2 times) e i 5 cπ/4; so that |0>|0>|0> has coefficient e i5·0π/4 = 1, |0>|0>|1> has coefficient e i5·1π/4 = –e iπ/4, |0>|1>|0> has coefficient e i5·2π/4 = i, etc. By a combination of entanglement (for the rows) and superposition (for the columns) this 6-qubit register now contains all 64 elements of the matrix e i a cπ/4, where a and c run from 0 to 7.
The Shor Factorization Algorithm
As set out in last month's column, we are working to discover the two prime factors of a number N by choosing an arbitrary number x smaller than N and detecting by Fourier analysis the periodicity of the sequence of remainders xa mod N, a= 0, 1, 2, etc.
We follow Shor, and work with a register L of length q, where N2 < 2q < 2N2, and a second register R large enough to hold N. Initially every qubit in L is in state |0>.
Step 1. We load into L an equiprobable superposition of all the numbers between 0 and 2q–1. This means applying the quantum gate Rj to qubitj, for j = 0, ... , q–1. These gates can be applied simultaneously.
Step 2. For each number a in L, we calculate xa mod N and write it in register R. This can be done simultaneously for all a in L. Now in registers L,R we have the superposition
Notation: Here and in what follows |k> in register L stands for |bq–1> ... |b1> |b0>, where bq–1 ... b1b0 is the binary representation of k, with a similar convention for register R.
Step 3. We apply the Quantum Fourier Transform to register L. This replaces each |a> by . In Fig. 1, each column in the matrix corresponds to a standard basis state for L, whereas the right-most column represents the contents of register R; the state of LR is the superposition
Fig. 1. Step 3 of the Shor Factorization Algorithm illustrated with N = 85, x = 19 and q = 4. Now the LR register holds a superposition of 162 states, each weighted by 1/16. Reading from the top left, the first one is e2 π 0·0/16|0> |1>, i.e.
the state corresponding to the last entry in the fourth row is e2 π 3·15/16|15> |59>, i.e.
Before going to Step 4, let us note that the registers are set up to carry out a Discrete Fourier Transform. But that algorithm would require 2q multiplications for each of 2q rows: many too many operations. Shor's intuition was that superposition and entanglement could be harnessed to do all the work in a couple of steps, by first reading register R and then reading register L.
Step 4. We examine register R. One of the values will appear (they all have equal probability), and the others will be lost. Suppose that value is xα mod N. During the readout the contents of register L collapse to those states which were coupled with xα mod N (compare Example 2 above).
Fig. 2. Step 4 of Shor's Factorization Algorithm illustrated with N = 85, x = 19 and q = 4. We suppose that register R reads 59 (so α, as above, was 3 or 11). Register L now only contains the superposition of states which were entangled with |59>; these are set off by the green boxes.
Fig. 3. Step 4 of Shor's Factorization Algorithm illustrated with N = 85, x = 33 and q = 4; conventions as above. We suppose that register R reads 67.
Step 5. We now read out register L. In the two examples we have considered, the readout will be an exact multiple of p = 2q/r. Repeating the experiment with the same x a small number of times leads with very high probability to a set of readouts whose only common divisor is p; so r can be determined and the problem is solved.
These examples are untypically simple in that r (8 and 4 in these two cases) was a power of 2, and therefore divided 2q exactly.
In fact, there will be exactly p = 2q/r values a in the range [0,2q–1] such that xa = xα mod N, and each of them will contribute
to register L. Since each of these a's is of the form α + kr, k = 0, ..., p–1, the factor of |c> is, up to a constant,
writing r = 2q/p for the last equality. The summation cycles through a symmetric set of p-th roots of 1, and is therefore zero, unless c is a multiple of p, in which case each term in the sum equals 1.
The general case requires a subtler analysis. Now 2q/r is not an integer. The readout in general cannot be an exact multiple of the frequency. But using a long enough sample (here is where the condition 2q > N2 comes into play --to analyze 85 "realistically" we would have had to use q = 13 and an 8192 x 8192 table of coefficients) guarantees, with high probability, a useful approximation. Complete details are in Shor's article referenced below.
The calculated probabilities of a readout of c, with r = 10 and 2q = 256. As Shor remarks, the value r = 10 could occur when factoring 33 if x were chosen to be 5, for example; 256 is chosen instead of the required 211 to get a legible picture. With high probability the observed value of c is near an integral multiple of 2q/r = 256/10. (Image courtesy Peter Shor.)
Peter W. Shor, Algorithms for Quantum Computation: In: Proceedings, 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, November 20--22, 1994, IEEE Computer Society Press, pp. 124--134.
History of the Quantum Fourier Transform:
R. Cleve, A note on computing Fourier transforms by quantum programs, unpublished, available as postscript file
D. Coppersmith, An Approximate Fourier Transform Useful in Quantum Factoring, IBM Research Report 07/12/94, available as arXiv:quant-ph/0201067.
A. Ekert and R. Jozsa, Quantum computation and Shor's factoring algorithm, Reviews of Modern Physics 68 (1996) 733-753
Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some these materials. Some of the items above can be accessed via the ACM Portal , which also provides bibliographic services.
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