Quantum Circuit Example Files

This directory contains the following quantum circuit (".qc") files. They can be loaded via web start or from within the jQuantum program from your hard disk. Both applications run if the Java Runtime Environment (JRE) by SUN is installed.

Survey

Classical Circuits
Non-classical Circuits
- Fourier-cNOT-12.qc
- Fourier-Toffoli-123.qc
- entireSuperpositionPhaseShifts.qc
- Fourier-cNOT-12.qc
- H-Fourier.qc
- Deutsch and Deutsch-Josza algorithm (Load: D00, D01, D10, D11, DJ0, DJ1, DJ-b-e, DJ-b-o)
- Shor-437.qc
- Grover-151.qc
Entanglement
- Bell states (Load: B00, B01, B10, B11)
- GHZ state (Load: GHZ, GHZ-A1, GHZ-A2, GHZ-A3, GHZ-B)
- W state

1. Classical Circuits

AND.qc - Quantum version of a classical AND gate. Here the third qubit x₃ always has the value x₁ x₂, where x₁ and x₂ are the respective values of the first and the second qubit:

x₁ x₂ x₃ ← x₁ x₂

0 0 0

0 1 0

1 0 0

1 1 1
OR.qc - Quantum version of a classical OR gate. Here the third qubit x₃ always has the value x₁ x₂, where x₁ and x₂ are the respective values of the first and the second qubit:

x₁ x₂ x₃ ← x₁ x₂

0 0 0

0 1 1

1 0 1

1 1 1
OR-unmodified.qc - Quantum circuit resulting in the same state for the third qubit |x₃> than the OR.qc circuit above, but now the input qubits |x₁> and |x₂> remain unchanged afterwards.
Adder.qc - Quantum version of a classical ADDER gate. Here the three qubits x₁, x₂ and x₃ are related by the following table:

x₁ x₂ x₃ ← carry = x₁ x₂ x₂ ← x₁ x₂

0 0 0 0

0 1 0 1

1 0 0 1

1 1 1 1

x₁	x₂	x₃ ← x₁ x₂
0	0	0
0	1	0
1	0	0
1	1	1

x₁	x₂	x₃ ← x₁ x₂
0	0	0
0	1	1
1	0	1
1	1	1

x₁	x₂	x₃ ← `carry` = x₁ x₂	x₂ ← x₁ x₂
0	0	0	0
0	1	0	1
1	0	0	1
1	1	1	1

2. Nonclassical Circuits

Fourier-cNOT-12.qc - An interesting simple quantum circuit consisting only of a quantum Fourier transform and a cNOT gate, starting with an initial state ≠ |0>.
Fourier-Toffoli-123.qc - Another interesting simple quantum circuit consisting only of a quantum Fourier transform and a Toffoli gate, starting with an initial state ≠ |0>.
entireSuperpositionPhaseShifts.qc - A quantum circuit consisting of a Hadamard, two sqrt-X gates a Pauli-Y gate and a classical cNOT, starting with an initial state ≠ |0> and resulting in an entirely superposed quantum register where the qubits are phased shifted with respect to each other.
H-Fourier.qc - A simple quantum circuit consisting of a Hadamard and quantum Fourier transform, starting with an initial state ≠ |0>.
Fourier-cNOT-12.qc - A simple quantum circuit consisting of a Hadamard, a cNOT and quantum Fourier transform, starting with an initial state ≠ |0>.
Deutsch and Deutsch-Josza algorithm - The Deutsch algorithm solves the problem: Given a black-box function f: {0, 1} → {0, 1} that maps one bit to one bit, determine whether the function is constant (i.e., f(0) = f(1) ) or balanced (i.e., f(0) ≠ f(1) ).
Classically we have to perform two computations of f to decide this problem. However, the Deutsch algorithm uses amplitude interferences of the superposed quantum states such that a single qubit measurement suffices to solve it. If the x qubit is 0, then the function is constant, otherwise it is balanced.

The files Deutsch-00.qc, Deutsch-01.qc, Deutsch-10.qc, Deutsch-11.qc represent the four possible cases of f which can occur.

The Deutsch-Josza algorithm solves the similar problem for a function f: {0, 1}ⁿ → {0, 1} which maps n bits to one bit with the promise that the function is constant (i.e., f(x) = f(y) ) or balanced (i.e., for each x there are exactly 2^n-1 different y such that f(x) ≠ f(y) ).

The files Deutsch-Josza-0.qc and Deutsch-Josza-1.qc correspond to the constant functions f(x) = 0 and f(x) = 1, respectively, whereas the other two circuits Deutsch-Josza-balanced-even.qc and Deutsch-Josza-balanced-odd.qc represent balanced functions where each even value gives 0 in one case, and each odd value gives 0 in the other case.
Shor-437.qc - The core of Shor's period finding quantum algorithm to find the prime factors of a given number. In the example, the number whose prime factors are searched is n=437, thus the implemented function is
f(x) = 4^x mod 437.

By the first Fourier gate, the entire x-register is set into a superposition, such that by the subsequent function evaluation gate the y-register attains all possible values f(x). A measurement of the y-register then yields all in x-values entangled with the measured y value, satisfying f(x) = y. After an inverse quantum Fourier transform of the x-register giving a superposition with sharp probability amplitude maxima at
k ≈ sq/p, s = 0, 1, 2, ..., p
the multiples of the searched for period p; q denotes the total number of possible register states of the x-register, i.e., q = 2ⁿ with n the size of the register. (The probability maxima are best viewed by enabling the "length coloring" option in the configuration menu.) A second measurement then yields one of these k-values with high probability. Knowing the register size q, we have thus in fact measured the ratio s/p (with high probabilty, at least). The classical method of continued fraction then yields a fraction whose denominator is a divisor of the searched for period p. Hence several repetitions of this procedure then enable to find the period p of f, which in turn gives the essential hint to determine the prime factors of n = 437 = 19 · 23.

For instance, a first measurement could give y = 301, which corresponds to the values x = 83, 182, 281, 380, 479, 578, 677, 776, 875, 974, ..., since the (searched for) period, or order, of 4 with respect to 301 is p = ord₃₀₁(4) = 99, cf. http://math-it.org/Mathematik/Zahlentheorie/OrdnungWert.html. Since the register contains 12 qubits, we have q = 2¹² = 4096 possible register states, and hence there are 41 ≈ q/p x-values stored in the x-register. Thus the inverse QFT yields a register with high probability amplitudes for multiples k = 41s of 41. This yields a measured value 41s, and the continued fraction algorithm yields 41s/4096 ≈ s/99, whose denominator is the searched for period or one of its divisors (in case s happens to divide 99).

Grover-151.qc - The Grover search algorithm applied to a quantum register of 8 qubits and using an oracle with the "needle" |x> = |151>. An oracle is a black box which can be asked whether a specified value equals the needle value or not, but which cannot give any more information about the needle. The Grover algorithm consists of applying a Fourier transform on the initial state |0> and r Grover operators, or Grover iteration steps, with

r &asymp

4 arcsin(2^-n/2)

≈

2^n/2 π

= 2^{(n - 4)/2} π

where n ≥ 2 is the size of the register, i.e., the number of its qubits. Each Grover iteration amplifies the searched value and weakens all other values of the register. It consists of the following steps:

Apply an oracle query which assigns +1 to each value which is different from the searched for value (the "needle"), and -1 to the value which is searched for.
Apply an n-fold Hadamard on the entire register
Apply the conditional phase shift of -1, -I₀, defined by -I₀(|0>) = |0> and
-I₀(|x>) = -|x> if x ≠ 0.
Apply an n-fold Hadamard on the entire register

The iteration step is described in M.A. Nielsen & I.L. Chuang: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000, §6.1.2.

In practice, an oracle could be a cryptologic algorithm in a known-plaintext attack which applies a secret key to a given ciphertext and yields -1 if the deciphered text equals the known plaintex, and +1 otherwise.

3. Entanglement

Bell-state-00.qc, Bell-state-01.qc, Bell-state-10.qc, Bell-state-11.qc - These quantum circuits construct the simplest entangled states, the Bell states or EPR states, consisting of two qubits in a superposition of two states which are not a product state. From the 8 possible states which two qubits can attain only the following four states are entangled:
$|\Phi^+\rangle = \frac{|00\rangle + |11\rangle}{\sqrt2}$ , $|\Psi^+\rangle = \frac{|01\rangle + |10\rangle}{\sqrt2}$ ,

$|\Phi^+\rangle = \frac{|00\rangle - |11\rangle}{\sqrt2}$ , $|\Psi^+\rangle = \frac{|01\rangle - |10\rangle}{\sqrt2}$ ,

The relevance of entangled states stems from the case that the entangled qubits are far apart from each other. A measurement of one of them then has an instant impact on the entangled qubit. In this way, entanglement can be used to perform teleportation or quantum communication.

GHZ.qc, GHZ-A1.qc, GHZ-A2.qc, GHZ-A3.qc, GHZ-B.qc -

The GHZ experiment, after Greenberger, Horne and Zeilinger, is a gedanken experiment in quantum mechanics which gives opposite results depending on whether quantum mechanics or Einstein's local realism with hidden variables holds. First published in 1989, it was performed in 1999 and falsified the predictions of hidden variable theories.

Whereas the Bell inequalities, which in essence decide the same question, require a statistical evaluation of large measurement series, the GHZ experiment only needs four measurements. Its core idea is to take three particles, each of which is a two-state quantum system attaining the states $|0\rangle$ and $|1\rangle$ , and entangle them into the GHZ state

$|\psi\rangle = \frac{|000\rangle + |111\rangle}{\sqrt2}$

Here $|0\rangle$ and $|1\rangle$ may represent the states spin-up and spin-down, for instance, with the eigenvalue +1 for $|0\rangle$ and -1 for $|1\rangle$ . As a quantum circuit, the GHZ state can be obtained from the initial state $|0\rangle$ $|0\rangle$ $|0\rangle$ by a Hadamard and two c-NOTs, cf. GHZ.qc: $|psi\rangle-as-circuit$ . With the first two Pauli matrices $X, Y$ we then construct the four measurement operators

A₁	=	X₁Y₂Y₃
A₂	=	Y₁X₂Y₃
A₃	=	Y₁Y₂X₃
B	=	X₁X₂X₃

Realized as a quantum register, a measurement in the Pauli basis X may be regarded as a transformation of the quantum state in the standard computational basis by H, i.e.,

$|0_x\rangle$ , $|1_x\rangle$ ,

or equivalently

$|0\rangle-in-x$ , $|1\rangle-in-x$ ,

and a subsequent measurement of the respective qubit; analogously, a measurement in the Pauli basis Y a transformation by S^†H

$|0_y\rangle$ , $|1_y\rangle$ ,

or equivalently

$|0\rangle-in-y$ , $|1\rangle-in-y$ ,

and a subsequent measurement. In both cases, the measurement result is λ = +1 if and only if the qubit state is $|0_x\rangle$ or $|0_y\rangle$ , respectively, and λ = -1 if and only if the qubit state is $|1_x\rangle$ or $|1_y\rangle$ , respectively. As can be seen with the quantum circuits GHZ-A1.qc, GHZ-A2.qc, GHZ-A3.qc, and GHZ-B.qc (or as can be directly verified by calculation), the qubits in the A_i-bases contain an odd number of 1-qubits,

$|psi\rangle-in-xyy$ ,

$|psi\rangle-in-yxy$ ,

$|psi\rangle-in-yyx$ ,

whereas in the B-basis we have only those qubits with even number of 1-qubits,

$|psi\rangle-in-xxx$ .

For each set up state, a measurement of two qubits suffices to determine the state of the third one uniquely. By construction, the measurement results are

A_{1, 2, 3} $\|\psi\rangle$	=	-1,
B $\|\psi\rangle$	=	+1.

But this last measurement result for B leads to a contradiction to Einstein's local realism: If the GHZ state $|psi\rangle$ was completely determined by some hidden variables and thus the same in each measurement trial, then we would have B $|\psi\rangle$ = -1, since Y₁Y₁ = Y₂Y₂ = Y₃Y₃ = +1 and therefore B = X₁X₁X₃ can be expressed as the product A₁A₂A₃,

A₁	=	X₁Y₂Y₃	=	-1
A₂	=	Y₁X₂Y₃	=	-1
A₃	=	Y₁Y₂X₃	=	-1

B	=	X₁X₂X₃	=	-1	(if local realism holds)

However, this is a contradiction to the result B $|\psi\rangle$ = +1 which quantum mechanics (without hidden variables) predicts. Realized GHZ experiments confirmed B $|\psi\rangle$ = +1 and therefore falsified local realism.

W-state.qc - Another interesting entangled state of three qubits, besides the GHZ state, is the W state consisting of a superposition of three states,

$|W\rangle$ .

It cannot be constructed neither from the state $|0\rangle$ nor from the GHZ state by two-product operations [W. Dür, G. Vidal, and J.I. Cirac, Phys. Rev. A 62, 062314 (2000)]. A way to create it is by means of function evaluation such that it is situated in the y-register.