Author: Anton Hibl, E-mail: [email protected]
The Quantum Playground package provides tools to quickly perform and test computations involving quantum compuational gates and their respective basis states. These are represented using Julia's already present Linear Algebra Syntax in unison with some of the higher level mathematical functions in the LinearAlgebra, Calculus, and Statistics packages. The goal is to create an easy to run environment in which the user can perform and test out small scale quantum computations and gate combinations to better understand the underlying concepts which motivate quantum computation. You are able to easily combine compuations utilizing different gates, basis states, rotations, measurements, and functions in an intuitive way. Due to the inclusion of Pluto the user may also use it inside of a notebook easily, which is useful for scientific documentation purposes.
The install process is undergoing construction as the package is still under construction, if you desire to test as is please clone the repository and follow the steps below to test the package:
- Ensure your shell is working inside of the directory you just cloned,
QuantumPlayground.jl. Start julia in your terminal using the
juliacommand. - type in
]activate . - type
updateto ensure all packages are updated - type
build - delete one char to exit pkg mode and type:
using QuantumPlayground; qp = QuantumPlayground; la = qp.LinearAlgebra
this set some shortcuts to qp and la so that these two short forms can be used to access the package instead of the longer words QuantumPlayground and LinearAlgebra every time we use them.
The package is now ready to use, type qp. and hit tab to see the list
of accessible tools.
Many of the standard gates for quantum computing are defined so that researchers may analyze and document how a quantum states, fields, and systems evolve over time. The following is the beginnings of this gate-set:
-
The NOT Gate(Pauli X) :
X = [0 1;1 0], Also called the negation, uses one input to generate one output. A NOT gate inverts the input. -
The Identity Gate :
I = [1 0;0 1], This is simply a gate that returns the identity of the matrix to which it is applied; it is also known as the identity matrix. -
The Hadamard Gate :
H = (1/√2)*[1 1;1 -1], A fundamental quantum gate which allows us to move away from the poles of the Bloch sphere and create a superposition of |0⟩ and |1⟩ -
The Pauli Y Gate :
Y = [0 -im;im 0], The Pauli Y gate is equivalent to Ry for the angle. It is equivalent to applying X and Z, up to a global phase factor. -
The Pauli Z Gate :
Z = [1 0;0 -1], The Pauli Z gate acts as identity on the state and multiplies the sign of the state by -1. It therefore flips the and states. In the +/- basis, it plays the same role as the NOT gate in the |0⟩/|1⟩ basis. -
The RX Gate :
RX = (1/√(2))*[1+im 1-im;1-im 1+im], The RX gate implements . On the Bloch sphere, this gate corresponds to rotating the qubit state around the x axis by the given angle. -
The Square Root of X Gate :
SX = (1/√(2))*[1 -im;-im 1], This gate implements the square-root of X, √X. Applying this gate twice in a row produces the standard Pauli-X gate. Like the Hadamard gate, creates an equal superposition state if the qubit is in the state , but with a different relative phase.
these can be used like so:
QuantumPlayground.<gate name here>
or
qp.<gate name here>
if aliased as mentioned above.
Common measurement operations can be performed in the same syntax albeit
calling as a function with parameters. Some of the measuring operations
are modlength, magnitude, and phase; modlength measures the length
of a complex number in the cartesian-complex plane by calculating
the modulus of the complex numbers components. The magnitude operation
measures a given statevector's magnitude or statevector length; a state
vector is typically a 2x1 or 1x2 vector constistant of 2 complex numbers
which represent the qubit's state. The phase operation measures the
phase of a complex number, in other words the angle which represents the
time on the polar plane; this is returned in terms of radians.
an example of a measurement involving an entire 4x4 matrix and the
modlength operation would look like:
qp.modlength.(qp.SX)
Plotting in various forms can be done in a few ways using a few different forms of complex numbers and accompanying differing representations to go alongside the respective forms. What is meant by this is that for the cartesian form of a complex number, a+bi, there is a cartesian plotting function to coincide; for the polar form of a complex number, (p, theta), there is also a coinciding polar plotting function to plot using the polar representation of the number. A bloch sphere/quaternion plotting function is in the works of being developed right now, stay tuned.
Some more complex gates, such as ones which perform series of rotations and phase modulations, are also available:
-
The S-Phase Gate :
S = [1 0;0 e^((im*pi)/2)], A quantum gate which can be utilized to initiate a phase of pi/2 in the state matrix it is applied towards. This essentially evaluates the state at a time pi/2. -
The T-Phase Gate :
T = [1 0;0 e^((im*pi)/4)], A quantum gate which can be utilized to initiate a phase of pi/4 in the state matrix it is applied towards. This essentially evaluates the state at a time pi/4. -
The Rotation Gate :
Rotation(matrix, theta), computes a rotation of the given matrix around the x-axis by a degrees of an angle theta. -
The U2-Rotational Gate :
U2_Rotation(matrix, theta, alpha), computes 2 consecutive rotations of a matrix; first around the x-axis according to degrees theta, then around the y-axis according to degrees alpha. -
The U3-Rotational Gate :
U3_Rotation(matrix, theta, alpha, beta), computes 3 consecutive rotations of a matrix around first the x axis according to degrees theta, then the y-axis according to degrees alpha, then finally around the z-axis according to degrees beta. -
The SZ Gate :
SZ = S*S, A quantum gate which performs a half-turn of what the usual Z-gate performs. -
The QZ Gate :
QZ = T*T*T*T, A quantum gate which performs a quarter-turn of what the usual Z-Gate performs. -
The Spinning Hadamard Gate :
SpinH = ((Rotation([1 1;0 0], pi))*H), The spinning hadamard gate represents a "spinning" superposition, wherein the randomness is more inherent than a typical H gate as it could now have rotated either direction. -
The Controlled NOT Gate :
CX = [1 0 0 0;0 1 0 0;0 0 0 1;0 0 1 0], the CX gate entangles 2 qubits, affecting the target qubit only if the control qubit is active or on; the target operation is a pauli-X Gate -
The Controlled-Y Gate :
CY = [1 0 0 0;0 1 0 0;0 0 0 -im;0 0 im 0], the CY gate entangles 2 qubits, affecting the target qubit only if the control qubit is active or on; the Target operation is a pauli-Y Gate. -
The Controlled-Z Gate :
CZ = [1 0 0 0;0 1 0 0;0 0 1 0;0 0 0 -1], The CZ gate entangles 2 qubits, affecting the target qubit only if the control qubit is active or on; the target operation is a pauli-Z Gate. -
The Toffoli Gate :
Toffoli = [1 0 0 0 0 0;0 1 0 0 0 0;0 0 1 0 0 0;0 0 0 1 0 0;0 0 0 0 0 1;0 0 0 0 1 0], The toffoli gate acts as an extension of the CX gate, now having 2 control qubits instead of just 1. -
The Fredkin Gate :
Fredkin = [1 0 0 0 0 0;0 1 0 0 0 0;0 0 1 0 0 0;0 0 0 01 0;0 0 0 1 0 0;0 0 0 0 0 1], The Fredkin Gate acts similarly to the Toffoli gate, having 2 control qubits, the main difference being the NOT acts between the 4th and 5th qubits.
Operations such as calculating the Hermitian can be performed too:
qp.Hermitian(qp.H)
this computes the hermitian(conjugate transpose) of a matrix, this should return the same matrix if it is unitary.
PhotonPlayground.jl serves as a sub-package to QuantumPlayground.jl where the user can explore the differences between classical qubit modeled quantum calculations and photonic continuous-variable modeled quantum calculations in terms of their matrix components and solving equations involving different quantum fields or systems.