JQ Manual

Quick Overview

For quick start, please note 3 inputs. We we will name them: 1st, 2nd and 3rd — from left to right. Please also note 2 buttons: Run and Load.

1st input - init state

You enter init state of qubits here. A decimal integer is required here. Its binary representation defines the start states of qubits in qregister. Qubit 0 is at the top.

2nd input - qubit signature

You enter a list of integers here in the following format: [qubit1, qubit2, ..., qubitn]. This qubit signature is used for displaying probabilities.

Assume you entered here [3,4] and assume that list of probabilities is like the following:

01 : 0.3 (1)
10 : 0.7 (2)

It simply means that a lot of hypothetical measurements would show that Prob(qubit 4 is in state 0 AND qubit 3 is in state 1) = 0.3 and Prob(qubit 4 is in state 1 AND qubit 3 is in state 0) = 0.7.

Formally, the result is a list of integers with a given probability. The n-th bit of the chosen integer (from the right) represents the state of the qubit which is listed as n-th in the qubit signature (from the left) after the hypothetical measurement. Note these integers are given in binary and in decimal in ().

The same result for the signature [4,3] would obviously be

01 : 0.7 (1)
10 : 0.3 (2)

3rd input - quantum circuit

You enter a list of quantum gates here in the following format [gate1, gate2, ..., gaten]. You can also add measurements to this list, they are represented by M(.).

Run Button

OK, after you've entered the initial state (if you don't, the default state is 0) and quantum circuit, you press the Run button, which applies the quantum circuit to the qregister and displays the end state and probabilities after a hypothetical measurement of the qubits listed in qubit signature (2nd input). For example:

01 : 0.5 (1)
10 : 0.5 (2)

(0.707106781187,0)|1>
(0.707106781187,0)|2>

The first list are probabilities described in 2nd input - qubit signature section. The second list is just a state of the qregister (a complex vector).

Load Button

When you create some more complex and long circuit, you want to have it in a saved text file. You can. Write your circuit in a text file using your favorite editor, save it on your drive and then just press Load to load it into the Quantum Simulator.

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Specification

Supported gates

H(bit) [Hadamard Gate]

Defined on base vectors $|[bit]\rangle$:

$H|0\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$

$H|1\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$

X(bit) [Pauli X-gate]

Defined on base vectors $|[bit]\rangle$:

$X|0\rangle = |1\rangle$

$X|1\rangle = |0\rangle$

Y(bit) [Pauli Y-gate]

Defined on base vectors $|[bit]\rangle$:

$X|0\rangle = i|1\rangle$

$X|1\rangle = -i|0\rangle$

Z(bit) [Pauli Z-gate]

Defined on base vectors $|[bit]\rangle$:

$X|0\rangle = |0\rangle$

$X|1\rangle = -|1\rangle$

R(n, bit) [Phase Shift Gate]

Defined on base vectors $|[bit]\rangle$:

$R_n|0\rangle = |0\rangle$

$R_n|1\rangle = e^{i\pi/2^n}|1\rangle$

CNot(control, bit) [Controled Not Gate]

Defined on base vectors $|[control][bit]\rangle$:

$CNot|00\rangle = |00\rangle$

$CNot|01\rangle = |01\rangle$

$CNot|10\rangle = |11\rangle$

$CNot|11\rangle = |10\rangle$

CCNot(control1, control2, bit) [Toffoli Gate]

Defined on base vectors $|[control1][control2][bit]\rangle$:

$CCNot(|c_1c_2b>) = |c_1c_2(c_1\cdot c_2\oplus b)\rangle$

C(control, gate) [Universal Control Gate]

Defined on base vectors $|[control]x\rangle$:

$C(gate)(|0\rangle|x\rangle) = |0\rangle|x\rangle$

$C(gate)(|1\rangle|x\rangle) = |0\rangle gate|x\rangle$

U([argbit₀, ...,argbit_n],[valbit₀, ...,valbit_m], function) [Universal Function Gate]

Defined on base vectors $|[argbit_n]\dots[argbit_0]\rangle|[valbit_m]\dots[valbit_0]\rangle$:

$U_{function}|x\rangle|y\rangle = |x\rangle|y \oplus function(x)\rangle$

Hidden Console

jQ has additional advanced feature — Hidden Console. When you place cursor in the last file of the black window in the left you can enter some special directives. Bellow you will find a full list of Hidden Console directives:

clr()

Clears window.

load()

The same as button Load.

measure(bit)

Performs measurement on chosen bit. To see result you need to type prob([[bit]]). Changes qregister state.

prob([signature₁, signature₂, ..., signature_n])

Displays states of chosen qubits with probabilities after series of hypothetical measurements. Doesn't change the qregister state. The value of state is binary grouped. Each group is defined by corresponding signature. Consider following example. Assume that the qregister state is:

(0.5,0)|0>
(0.5,0)|1>
(0.5,0)|4>
(0.5,0)|5>

Corresponding probabilities are:

000 : 0.25 (0)
001 : 0.25 (1)
100 : 0.25 (4)
101 : 0.25 (5)

After typing in Hidden Console prob( [ [0], [1,2] ] )

You will see the same result in different representation:

|0>|0>:0.25
|0>|1>:0.25
|2>|0>:0.25
|2>|1>:0.25

For prob( [ [0,1], [2] ]):

|0>|0>:0.25
|0>|1>:0.25
|1>|0>:0.25
|1>|1>:0.25

reset(n)

Set initial state of qregister to n.

run()

The same as button Run.

state()

Displays current state (complex vector) of qregister.