The mathematics behind quantum computing

The second of two feature columns on this subject.

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.

Quantum computers

Data in a quantum computer are stored in qubits, and manipulated by gates.

• A qubit (the name is a contraction of "quantum bit") is a device whose state can be represented by a unit vector in a 2-dimensional complex vector space. In terms of an orthonormal basis, usually designated |0>, |1>, the state is a₀|0> + a₁|1>; here a₀ and a₁ are complex numbers satisfying |a₀|² + |a₁|² = 1. When the qubit is measured, it reports "0" with probability |a₀|² and "1" with probability |a₁|²; meanwhile, the numbers a₀ and a₁ are lost.
  
• Gates have kept their name from an early mechanistic conception of the flow of information through a computer. In quantum computation, gates are physical processes which can be applied to qubits.
  • One-qubit gates. For the operation to be realizable by a quantum-mechanical process, the output vector (b₀, b₁) must also be a unit vector, and it must vary linearly with the input vector. In other words, if V is the vector space of states of the qubit in question, the gate must correspond to a unitary operator on V.

    Example: the NOT-gate. This is the simplest example of a gate that does something.

    NOT-gate:

    INPUT OUTPUT |0> |1> |1> |0>

    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:

    R-gate:

    INPUT OUTPUT |0> |0> + |1> |1> |0> – |1>

    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.

  • Two-qubit gates. Now, for physical realizability, each output state must vary linearly with each of the input states. This means that the state space of a pair of qubits is the tensor product of their individual state spaces, and that the gate is represented by an invertible linear operator on that tensor product.

    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:

    (

    a0b0   a0b1 a1b0   a1b1

    ).

    In terms of the bases |0>, |1> and |0>, |1>, the tensor products

    |0> |0> = ( 1   0 0   0 ),	|0> |1> = ( 0   1 0   0 ),
    |1> |0> = ( 0   0 1   0 ),	|1> |1> = ( 0   0 0   1 )

    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>.

    C-NOT gate:

    INPUT OUTPUT    |0>|0>    |0>|1>    |1>|0>    |1>|1>    |0>|0>    |0>|1>    |1>|1>    |1>|0>

    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.

Entanglement

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:

c_0,0|0> |0>  +   c_0,1|0> |1>  +   c_1,0|1> |0>  +   c_1,1|1> |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 c_0,0 c_1,1 – c_1,0 c_0,1 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:

• Example of entanglement. Suppose a 2-qubit register is in the state |0> |0> + |1> |1>. When measured, this register has probability 1/2 of being in the state |0> |0>, and probability 1/2 of being in the state |1> |1>. If qubit is measured, and found to be in the state |0>, then when qubit is measured it must also be in the state |0>. It was entangled. Note that in the non-entangled case, measurement of qubit gives no information about the state of qubit.

Parallel processing in a quantum computer

• Example 1. Addition modulo 2. This example is elementary but illustrative. We work with a 3-qubit register
  qubit   qubit   qubit

  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

  D gate:

  INPUT     OUTPUT |0> |0> |0>     |0> |0> |0> |0> |0> |1>     |0> |0> |1> |0> |1> |0>     |0> |1> |1> |0> |1> |1>     |0> |1> |0> |1> |0> |0>     |1> |0> |1> |1> |0> |1>     |1> |0> |0> |1> |1> |0>     |1> |1> |0> |1> |1> |1>     |1> |1> |1>.

  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

  (1/2) |0> |0> |0> + (1/2) |0> |1> |0> + (1/2) |1> |0> |0> + (1/2) |1> |1> |0>.

  We now apply the D-gate once to this register. The output is

  (1/2) |0> |0> |0> + (1/2) |0> |1> |1> + (1/2) |1> |0> |1> + (1/2) |1> |1> |0>.

  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.
     All of the sums have been carried out at once! But this is not completely useful, since if we wanted the program to calculate the sum of a specific pair, we would need to continue loading and running until that pair came up in the input. Part of the subtlety of quantum programming is to structure algorithms so that the desired information can be extracted with high probability in a small number of runs.

• Example 2. Another way of reading the output from our experiment with the D-gate is to measure the output qubit. This measurement will yield 0 or 1 with probability 1/2. Suppose it reads 0. Then if the first two qubits are measured, they will report 0,0 or 1,1 with equal probability. The first measurement has collapsed the state space of the register to allow only these two possibilities. In other words, it now only contains those pairs of states whose mod-2 sum is zero.

• More generally, working with an n-qubit register, n applications of the R-gate (which can be carried out simultaneously) will produce an equiprobable superposition of the binary forms of all the numbers from 0 to 2ⁿ-1. Any operation that can be encoded in binary arithmetic can be carried out on all the numbers simultaneously. Typically the number of steps in this operation is a low-degree polynomial function of n, the number of binary places. This contrast between an exponential range and a polynomial number of steps is the key to the efficiency of quantum computation.

The Quantum Fourier Transform

• Example: the QFT on 3 bits.

  For 3 bits, the QFT algorithm specifies the sequence
  R₂ S_1,2 R₁ S_0,2 S_0,1 R₀
  applied in left-to-right order. Here we abbreviate |0>|0>|0> as 000, etc.

  input   000   001   010   011   100   101   110   111

  after R2 2–1/2 (000 + 001) 2–1/2 (000 – 001) 2–1/2 (010 + 011) 2–1/2 (010 – 011) 2–1/2 (100 + 101) 2–1/2 (100 – 101) 2–1/2 (110 + 111) 2–1/2 (110 – 111)

  after S1,2 (ω2–1=i) 2–1/2 (000 + 001) 2–1/2 (000 – 001) 2–1/2 (010 +i 011) 2–1/2 (010 –i 011) 2–1/2 (100 + 101) 2–1/2 (100 – 101) 2–1/2 (110 + i111) 2–1/2 (110 – i111)

  after R1 2–1 (000 + 010 + 001 + 011) 2–1 (000 + 010 – 001 – 011) 2–1 (000 – 010 + i001 – i011) 2–1 (000 – 010 – i001 + 011 ) 2–1 (100 + 110 + 101 + 111) 2–1 (100 + 110 – 101 – 111) 2–1 (100 – 110 + i101 – i111) 2–1 (100 – 110 – i101 + i111)

  after S0,2 (ω2–0=e iπ/4 ) 2–1 (000 + 010 + 001 + 011) 2–1 (000 + 010 – 001 – 011) 2–1 (000 – 010 + i001 – i011) 2–1 (000 – 010 – i001 + i011 ) 2–1 (100 + 110 + e iπ/4 101 + e iπ/4 111) 2–1 (100 + 110 – e iπ/4 101 – e iπ/4 111) 2–1 (100 – 110 + ie iπ/4 101 – ie iπ/4 111) 2–1 (100 – 110 – ie iπ/4 101 + ie iπ/4 111)

  after S0,1 (ω1–0=i) 2–1 (000 + 010 + 001 + 011) 2–1 (000 + 010 – 001 – 011) 2–1 (000 – 010 + i001 – i011) 2–1 (000 – 010 – i001 + i011 ) 2–1 (100 + i110 + e iπ/4 101 + ie iπ/4 111) 2–1 (100 + i110 – e iπ/4 101 – ie iπ/4 111) 2–1 (100 – i110 + ie iπ/4 101 + e iπ/4 111) 2–1 (100 – i110 – ie iπ/4 101 – e iπ/4 111)

  after R0 2–3/2 (000 + 100 + 010 + 110 + 001 + 101 + 011 + 111) 2–3/2 (000 + 100 + 010 + 110 – 001 – 101 – 011 – 111) 2–3/2 (000 + 100 – 010 – 110 + i001 + i101 – i011 –i111) 2–3/2 (000 + 100 – 010 – 110 – i001 – i101 + i011 + i111) 2–3/2 (000 – 100 + i010 – i110 + e iπ/4 001 – e iπ/4 101 + ie iπ/4 011 – ie iπ/4 111) 2–3/2 (000 – 100 + i010 – i110 – e iπ/4 001 + e iπ/4 101 – ie iπ/4 011 + ie iπ/4 111) 2–3/2 (000 – 100 – i010 + i110 + ie iπ/4 001 – ie iπ/4 101 + e iπ/4 011 – e iπ/4 111) 2–3/2 (000 – 100 – i010 + i110 – ie iπ/4 001 + ie iπ/4 101 – e iπ/4 011 + e iπ/4 111)

• For 4 bits the QFT algorithm would read R₃ S_2,3 R₂ S_1,3 S_1,2 R₁ S_0,3 S_0,2 S_0,1 R₀; in general for n bits the Quantum Fourier Transform requires (n² + n)/2 operations.

• Note that the Quantum Fourier Transform is not transforming anything. It gives a way to duplicate through quantum operations the steps of the Fast Fourier Transform, but without an input sequence. If the QFT is applied to the equiprobable superposition of the binary forms of the numbers from 0 to 2ⁿ-1, then in (n² + n)/2 steps it will produce a superposition of all the 2ⁿ rows of the matrix with a,c entry e^{a c i π/2n-1}.

  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>
            + e ^iπ/4 |1>|0>|1> – ie ^iπ/4 |0>|1>|1> + ie ^iπ/4 |1>|1>|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 x^a mod N, a= 0, 1, 2, etc.

We follow Shor, and work with a register L of length q, where N² < 2^q < 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 2^q–1. This means applying the quantum gate R_j to qubit_j, for j = 0, ... , q–1. These gates can be applied simultaneously.

Step 2. For each number a in L, we calculate x^a 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 |b_q–1> ... |b₁> |b₀>, where b_q–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 e^{2 π 0·0/16}|0> |1>, i.e.

(e^{2 π 0·0/16} |0> |0> |0> |0>) (|0> |0> |0> |0> |0> |0> |1>);

the state corresponding to the last entry in the fourth row is e^{2 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 2^q multiplications for each of 2^q 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 x^α mod N. During the readout the contents of register L collapse to those states which were coupled with x^α 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 = 2^q/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 2^q exactly.

• In this simple situation, we can see how the readout takes us from a superposition of states (in register R) with period r to a superposition of states (in register L) which have coefficient zero unless they correspond to a multiple of 2^q/r. (Compare the examples; this is the way the QFT detects frequency).

  In fact, there will be exactly p = 2^q/r values a in the range [0,2^q–1] such that x^a = x^α 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 = 2^q/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 2^q/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 2^q > 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 2^q = 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 2^q/r = 256/10. Image courtesy Peter Shor.

References

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.
An expanded version is available under the title
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, as arXiv:quant-ph/9508027.

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