Gate Pallet

This section references the various classical and quantum operations you can use to manipulate qubits in the quantum circuit using the gates. Quantum operations include quantum gates, such as the Hadamard gate, but not quantum gates, such as the measurement operation, barrier, reset, QFT, and iQFT block, etc.

Today’s computers—in theory (Turing machines) and practice (PCs, HPCs, laptops, tablets, smartphones,)—are based on classical physics. They are limited by locality (operations have only local effects) and by the classical fact that systems can be in only one state at a time. However, modern quantum physics tells us that the world behaves quite differently. A quantum system can simultaneously be in a superposition of many different states and exhibit interference effects during its evolution. Moreover, spatially separated quantum systems may be entangled, and operations may have “non-local” effects.

Before we proceed to the Quantum gates for basic understanding we should know about some basic concepts of Quantum-like :

Quantum Superposition
Quantum Entanglement
Quantum Interference

Let’s dive more into these concepts

Quantum Superposition

Quantum superposition is one of the fundamental principles of quantum mechanics. It states that a quantum system can exist in multiple states simultaneously until it is measured. Mathematically, it is represented as a linear combination of the basis states of the system.

Mathematical representation:

Let |ψ⟩ be the state vector of a quantum system. In a basis { |1⟩, |2⟩, …, |n⟩ }, the state vector can be written as:

|ψ⟩ = c₁|1⟩ + c₂|2⟩ + … + cn|n⟩ where c₁, c₂, …, cn are complex numbers called probability amplitudes.

The probability of measuring the system in state |i⟩ is given by |ci|².

Mathematical Formalism

Hilbert Space: The state of a quantum system is represented as a vector in a complex Hilbert space. This space is infinite-dimensional for systems with continuous degrees of freedom (like position or momentum) and finite-dimensional for systems with discrete degrees of freedom (like spin).
Basis States: A basis for this Hilbert space is a set of orthogonal, normalized vectors that span the space. These vectors represent the possible states of the system.
Linear Combination: A quantum system can be in a superposition of multiple basis states. This is represented mathematically as a linear combination of these states:

|ψ⟩ = c₁|ψ₁⟩ + c₂|ψ₂⟩ + ... + cn|ψn⟩

Here, |ψ⟩ is the state vector of the system, |ψ₁, |ψ₂, …, |ψn⟩ are the basis states, and c₁, c₂, …, cn are complex coefficients. The coefficients represent the amplitudes of the corresponding basis states.

Probability Amplitudes: The square of the absolute value of a coefficient, |ci|², gives the probability of finding the system in the corresponding basis state |ψi⟩ upon measurement.

Theoretical explanation:

Quantum superposition arises from the wave-particle duality of matter. At the quantum level, particles behave both as waves and particles. The wave nature of particles allows them to interfere with each other, which leads to superposition.

When a quantum system is prepared in a superposition state, it is said to be in a coherent state. This means that the different states that the system can be in are correlated with each other. As long as the system remains undisturbed, it will continue to be in a superposition state.

However, when the system is measured, the superposition collapses into one of the possible states. This is called the measurement problem in quantum mechanics. The outcome of the measurement is probabilistic and cannot be predicted with certainty.

Applications of quantum superposition:

Quantum superposition has many applications in quantum computing and quantum communication. In quantum computing, it is used to create qubits, which are the basic units of quantum information. Qubits can exist in superposition states, which allows them to perform calculations much faster than classical computers.

In quantum communication, quantum superposition is used to create quantum cryptography, which is a method of secure communication that is impossible to eavesdrop on. Quantum superposition is also used in quantum teleportation, which allows information to be transmitted instantly over long distances.

Some important points to remember about quantum superposition:

Quantum superposition is a fundamental principle of quantum mechanics.
It is represented mathematically as a linear combination of the basis states of a quantum system.
The probability of measuring a system in a particular state is given by the square of the magnitude of the corresponding probability amplitude.
Quantum superposition is a consequence of the wave-particle duality of matter.
It is used in quantum computing and quantum communication.

Quantum Entanglement

Quantum entanglement is a phenomenon in quantum mechanics where two or more particles become intrinsically linked, regardless of the distance between them. Once entangled, the state of one particle cannot be described independently of the state of the other(s). If a measurement is made on one particle, the state of the other particle is instantaneously affected.

Mathematical representation:

Quantum entanglement is mathematically represented using the concept of tensor products. Let |ψ₁⟩ and |ψ₂⟩ be the state vectors of two particles, respectively. The entangled state of the two particles can be written as:

|ψ⟩ = α|ψ₁⟩ ⊗ |ψ₂⟩ + β|ψ₁’⟩ ⊗ |ψ₂’⟩

where α and β are complex numbers, and |ψ₁’⟩ and |ψ₂’⟩ are different states of the respective particles. The key point here is that the state of the entangled system cannot be factored into the product of the individual states of the particles.

Theoretical explanation:

Quantum entanglement arises from the non-local nature of quantum mechanics. Unlike classical physics, where particles have definite properties, quantum particles can exist in multiple states simultaneously until they are measured. When two particles become entangled, their states become correlated in a way that cannot be explained by classical physics.

The measurement of one particle collapses the wave function of the entangled system, instantaneously affecting the state of the other particle, regardless of the distance between them. This phenomenon is known as non-local action and is one of the most counterintuitive aspects of quantum mechanics.

Applications of quantum entanglement:

Quantum entanglement has many potential applications in various fields, including:

Quantum computing: Entangled particles can be used to perform quantum computations much faster than classical computers.
Quantum cryptography: Entanglement can be used to create unbreakable encryption schemes.
Quantum teleportation: Information can be transmitted instantaneously over long distances using entangled particles.
Quantum sensing: Entanglement can be used to create highly sensitive sensors for various applications.

Some important points to remember about quantum entanglement:

Quantum entanglement is a non-local phenomenon that cannot be explained by classical physics.
It is represented mathematically using the concept of tensor products.
The measurement of one entangled particle instantaneously affects the state of the other particle, regardless of the distance between them.
Quantum entanglement has potential applications in various fields, including quantum computing, quantum cryptography, quantum teleportation, and quantum sensing.

Quantum Interference

Quantum interference is a phenomenon in quantum mechanics where two or more quantum states can combine to produce a new state with different properties. This is analogous to the interference of waves in classical physics, where two waves can combine to produce a wave with a larger or smaller amplitude.

Mathematical representation:

Let |ψ₁⟩ and |ψ₂⟩ be two quantum states. The superposition of these states can be written as:

|ψ⟩ = α|ψ₁⟩ + β|ψ₂⟩

where α and β are complex numbers. The probability of measuring the system in state |ψ₁⟩ or |ψ₂⟩ is given by |α|² and |β|², respectively.

The interference term between the two states is given by αβ*. If αβ* is positive, the interference is constructive, and the probability of measuring the system in state |ψ⟩ is larger than the sum of the probabilities of measuring the system in states |ψ₁⟩ and |ψ₂⟩. If αβ* is negative, the interference is destructive, and the probability of measuring the system in state |ψ⟩ is smaller than the sum of the probabilities of measuring the system in states |ψ₁⟩ and |ψ₂⟩.

Theoretical explanation:

Quantum interference arises from the wave-particle duality of matter. At the quantum level, particles behave both as waves and particles. The wave nature of particles allows them to interfere with each other, which leads to quantum interference.

When two quantum states are combined, their wave functions can interfere constructively or destructively. Constructive interference leads to an increase in the probability of measuring the system in the combined state, while destructive interference leads to a decrease in the probability.

Applications of quantum interference:

Quantum interference has many applications in various fields, including:

Quantum computing: Quantum interference is used to create qubits, which are the basic units of quantum information. Qubits can exist in superposition states, which allows them to perform calculations much faster than classical computers.
Quantum cryptography: Quantum interference is used to create quantum cryptography, which is a method of secure communication that is impossible to eavesdrop on.
Quantum sensing: Quantum interference is used to create highly sensitive sensors for various applications.

Some important points to remember about quantum interference:

Quantum interference is a phenomenon in quantum mechanics where two or more quantum states can combine to produce a new state with different properties.
It is represented mathematically using the concept of superposition and interference terms.
Quantum interference arises from the wave-particle duality of matter.
It has applications in various fields, including quantum computing, quantum cryptography, and quantum sensing.

Gates pallet:

Each entry below provides details, gate matrix reference, and the OpenQASM reference for each gate operation.

In the field of quantum computing and the quantum circuit model of computation, a quantum logic gate (or quantum gate) is a fundamental building block that operates on qubits. Quantum gates take advantage of two key aspects of quantum mechanics, superposition, and entanglement, which are inaccessible to classical gates. You can drag these gates and other operations onto the graphical circuit editor/composer. The different types of gates are color-coded, with classical gates in dark blue, phase gates in light blue, and non-unitary operations in grey. Click on a gate and select “Info (i)” to learn more about its definition or associated reference.

Also presented here is a short video tutorial on using these gates and their respective functionalities.

Hadamard Gate(H gate)

H Gate Matrix

The H, or Hadamard, gate that rotates the states is useful for making superpositions. If you have a universal gate set on a classical computer and add the Hadamard gate, it becomes a universal gate set on a quantum computer.

Open QASM reference code: h q[0];

Gate Matrix reference:

X-Axis Y-Axis Z-Axis

Y gate

Y-Gate Matrix

The Pauli Y gate is equivalent to Ry for the angle Π. It is equivalent to applying X and Z, to a global phase factor.

Open QASM reference code: y q[0];

Gate Matrix reference:

X-Axis Y-Axis Z-Axis

Z gate

Z-Gate Matrix

The Pauli Z gate acts as identity on the ∣0⟩ state and multiplies the sign of the ∣1⟩ 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.

Open QASM reference code: z q[0];

Gate Matrix reference:

X-Axis Y-Axis Z-Axis

T gate

T-Gate Matrix

The T gate is equivalent to RZ for the angle π/4. In quantum computing, the T gate is a single-qubit gate that creates phase shifts. It’s also known as the π/4 gate.

The T gate rotates the state of a qubit by an angle of π/4 radians. It applies a phase shift to the qubit’s state, specifically to the |1> state, while leaving the |0> state unchanged.
The T gate is also known as the π/8 gate because of how the RZ(π/4) matrix looks like. It’s also known as the fourth root of the Pauli Z gate because applying the T gate four times is equivalent to applying the Pauli Z gate.

Open QASM reference code: t q[0];

Gate Matrix reference:

X-Axis Y-Axis Z-Axis

S gate

S-Gate Matrix

The S gate applies a phase of i to the ∣1⟩ state. It is equivalent to RZ for the angle π/2. Also S=P(π/2). Referring to quantum mechanics, the S gate is a 90-degree rotation around the z-axis. It is also known as the phase gate or the Z90 gate. The S gate is related to the T gate by the relationship S = T 2

Open QASM reference code: s q[0];

Gate Matrix reference:

X-Axis Y-Axis Z-Axis

P Gate

P(Phase) Gate Matrix

The phase gate (P-gate) is a gate. It is also known as the general phase shift operator in quantum computing.

In Qiskit, the phase gate can be applied to any qubit by calling the p() method on the Quantum Circuit. The p() method is parameterized, so the phase (in radians) that needs to be applied to the qubit must be passed to it.

Open QASM reference code: p(theta) q[0];

$T^{†}$ gate

$T^{†}$ gate Matrix

In quantum computing, the T gate is a single-qubit gate that creates phase shifts. It rotates the state of a qubit by an angle of π/4 radians. Also known as the Tdg or T-dagger gate. The T gate is also known as the π/4 gate. It is related to the S gate by the relationship S = T 2.

Open QASM reference code: tdg q[0];

$S^{†}$ gate

$S^{†}$ gate Matrix

Also known as the Sdg or S-dagger gate. The inverse of the S gate.

Open QASM reference code: sdg q[0];

RX gate

RX Gate Matrix

In quantum computing, an RX gate is a single-qubit rotation operator that rotates a qubit around the x-axis of the Bloch Sphere by a specified angle, usually denoted by θ. The angle of rotation can be positive or negative and must be specified in radians. The RX gate is represented by the matrix RX, where θ is the angle of rotation in radians in the counter-clockwise direction.

The RX gate is a crucial component in many quantum algorithms and protocols due to its ability to rotate quantum states around the x-axis, making it a valuable tool for manipulating qubits in quantum computing.

In Qiskit, the RX gate can be easily implemented using the following function:

qc.rx(pi, q[0]). Here, “pi” represents the rotation amount, and “q[0]” denotes the qubit to which we want to apply the RX gate.

RY gate

RY Gate Matrix

In quantum computing, a Ry gate is a single-qubit rotation operator that rotates a qubit around the y-axis by a specified angle, usually denoted by θ. The angle of rotation can be positive or negative and must be specified in radians.

In Qiskit, we can easily implement an RY gate using the following function:

ry(pi, q[0]), where pi is the rotation amount and q[0] is the target qubit for the RY gate.

RZ gate

RZ Gate Matrix, where ϕ = θ/2

The Rz-gate or the Rϕ-gate would be the first parametrized gate that will be introduced. By “parametrized” we mean that it accepts a parameter and performs an operation based on this parameter. The parameter accepted here is ϕ and the operation performed is rotation around the z-axis by ϕ radians.

NOT Gate(Pauli-X)

Pauli -X Gate Pauli -Y Gate Pauli -Z Gate

The NOT gate, also known as the Pauli X gate, flips the ∣0⟩ state to ∣1⟩, and vice versa. The NOT gate is equivalent to RX for the angle π or ‘HZH’.The quantum NOT gate is a fundamental quantum gate that plays a crucial role in quantum computing. It is the analog of the classical NOT gate and can be represented in a quantum circuit as a circle with a cross in it.

The NOT gate is reversible, meaning that if you apply a NOT gate twice to the same signal, you get out the same value you started with. The NOT gate is also known as an inverter because it gives a negative mathematical output.

Open QASM reference code : x q[0];

Gate Matrix reference:

X-Axis Y-Axis Z-Axis

CNOT Gate(CX)

CNOT Gate Matrix

The controlled-NOT gate is also called the controlled-x (CX) gate. It operates on a pair of qubits, where one qubit acts as the ‘control’ and the other as the ‘target’. It executes a NOT operation on the target qubit if the control qubit is in the state ∣1.⟩

If the control qubit is in a superposition, this gate creates entanglement.

Open QASM reference code : cx q[0], q[1];

Toffoli gate(CCX)

Toffoli Gate Matrix

The Toffoli gate, also known as the double controlled-NOT gate (CCX), has two control qubits and one target. It applies a NOT to the target only when both controls are in state |1〉. Toffoli gate with the Hadamard gate is a universal gate set for quantum computing.

Open QASM reference code : ccx q[0], q[1], q[2];

Identity gate

Identity Gate Matrix

The I-gate does not do anything in particular. Its matrix is the Identity matrix itself. It is considered a gate because they are often useful in calculations. Also, they are significant in a way that they can be used to specify a “none” operation when considering real hardware. The identity gate is often used in quantum circuits for various purposes, such as padding or preparing qubits for subsequent operations without altering their states.

Open QASM reference code: id q[0];

Gate Matrix reference:

X-Axis Y-Axis Z-Axis

SWAP gate

Swap Gate Matrix

The Swap gate, often denoted as SWAP, is a two-qubit quantum gate used in quantum computing. As the name suggests, it swaps the states of two qubits.

Measurement Gate

Measurement in the standard basis, also known as the z basis or computational basis. Can be used to implement any kind of measurement when combined with gates. It is not a reversible operation. Quantum measurements are typically described as operations that collapse the quantum state to one of its basis states with certain probabilities.

In a quantum circuit, a measurement is represented by a measurement operator or a measurement gate. The most common type of measurement is the projective measurement, which projects the quantum state onto one of the basis states. The measurement outcomes are probabilistic, and the probabilities are determined by the magnitudes of the probability amplitudes associated with each basis state.

The measurement process is often denoted by the symbol M in quantum circuits.

Reset Gate

In quantum computing, the reset gate, also known as the initialization gate, doesn’t have a single, universally accepted mathematical expression. This is because it’s not a fundamental unitary operation like the RX, RY, or RZ gates. It’s a conceptual operation achieved through a combination of techniques. Here’s a breakdown of the reset gate and its implementation:

Function:

The reset gate aims to bring a qubit to the ground state, which is typically represented as |0⟩ in the computational basis.
However, unlike classical bits that can be directly set to 0 or 1, qubits exist in superpositions. Resetting a qubit requires manipulating its state to favor the |0⟩ basis.

Implementation Approaches:

There are two common approaches to implement a reset gate:

Measurement and Conditional Operation:

- This approach involves performing a measurement on the qubit in the Z-basis (|0⟩ or |1⟩).
- If the measurement outcome is |1⟩, a bit-flip operation (X gate) is applied to the qubit to flip it to the |0⟩ state. If the outcome is already |0⟩, no further operation is needed.

Error Correction Techniques (Advanced):

- - In practice, real quantum hardware can introduce errors during operations.
  - Advanced techniques from the field of quantum error correction can be used to “reset” a qubit by manipulating its state to be closer to the ideal |0⟩ state while mitigating potential errors.

Expressions for Implementation:

The specific expressions for the reset operation depend on the chosen approach.
- Measurement-based approach:
  - Measurement operator (in Z-basis): |Z⟩⟨Z| (projects onto the |0⟩ state) or |X⟩⟨X| (projects onto the |1⟩ state).
  - Conditional X gate: X (applied only if measurement outcome is |1⟩).
- Error correction techniques: These involve complex mathematical formulations specific to the chosen error correction code.

Barrier operation

In quantum computing, a barrier is a visualization tool used in quantum circuit diagrams. The barrier is not an actual quantum gate or operation but is employed to separate and group quantum gates within a circuit. It is represented by a double vertical line.

The barrier helps to organize and clarify the structure of a quantum circuit, especially when dealing with complex algorithms or when trying to highlight certain parts of the circuit for analysis. It does not affect the quantum computation itself and is purely a visual aid.

By inserting a barrier in a quantum circuit diagram, one can emphasize specific sections, making it easier to understand the logical structure of the algorithm or to isolate particular quantum operations. It is particularly useful in cases where different parts of the circuit are functionally distinct or are intended to be treated as separate entities.

Quantum Fourier Transform(QFT)

QFT(Expanded view)

The Quantum Fourier Transform (QFT) is a fundamental operation in quantum computing that plays a pivotal role in various quantum algorithms, particularly those related to quantum cryptography and quantum simulation. It extends the classical Fourier Transform to quantum mechanics, enabling the manipulation of information encoded in the amplitudes of quantum states.

QFT transforms quantum states from one basis to another, specifically between the computational (Z) basis and the Fourier basis. While classical Fourier Transform operates on continuous functions, QFT operates on the amplitudes of quantum states, which can represent multiple possibilities simultaneously due to superposition.

In quantum circuits, the Hadamard gate (H-gate) acts as the single-qubit QFT, converting between the Z-basis and X-basis states. The QFT is also applicable to multi-qubit systems, where it transforms the entire quantum state from the computational basis to the Fourier basis and vice versa.

One of the most prominent applications of QFT is in Shor’s algorithm, a quantum algorithm for integer factorization, where it enables efficient computation of the discrete Fourier transform, a crucial step in the algorithm’s implementation. Additionally, QFT finds applications in quantum phase estimation, quantum error correction, and various quantum machine learning algorithms. Despite its significance, the implementation of QFT in large-scale quantum systems remains a challenge due to the sensitivity of quantum states to errors and decoherence.

Step 1: Apply a Hadamard gate (H-gate) to the first qubit.

Apply controlled rotations (phase gates) on subsequent qubits, controlled by previous qubits.

Step 2: Controlled Rotations:

For each qubit j from 2 to n:

Apply Hadamard gate (H-gate) to qubit j.
Apply controlled rotations (phase gates) on qubit j, controlled by qubits 1 to j-1.
Our phases will increase π/2**1, π/2**2, till the number of qubits j-1.

Step 3: Swap Gates

Perform swap gates to reverse the order of the qubits.

Inverse Quantum Fourier Transform(iQFT)

iQFT(Expanded view)

The Inverse Quantum Fourier Transform (IQFT) is a crucial tool in quantum computing, particularly in quantum algorithms such as Shor’s algorithm and quantum phase estimation. It is the opposite operation of the Quantum Fourier Transform (QFT), which is analogous to the classical Discrete Fourier Transform (DFT).

Here’s a brief overview of the iQFT and its significance in quantum computing:

Purpose: The IQFT is used to convert quantum states from the frequency domain back to the time domain. In other words, it reverses the effect of the QFT, allowing us to retrieve the original quantum state after it has been transformed.

Algorithmic Applications: The QFT is applied to a quantum state representing the period of a function in Shor’s algorithm. After performing operations on this transformed state, the IQFT is used to recover the period, which is crucial for factoring large numbers efficiently. Quantum phase estimation is another algorithm where the QFT and IQFT play a significant role. The QFT is applied to the eigenvectors of a unitary operator to estimate their corresponding eigenvalues. The IQFT is then applied to convert these estimated eigenvalues back into the original quantum states.

Implementation: The implementation of the IQFT depends on the specific quantum architecture and the available quantum gates. Like the QFT, the IQFT is typically implemented using a combination of Hadamard gates and controlled-phase gates.

Efficiency: The efficiency of the IQFT, like other quantum algorithms, depends on factors such as the number of qubits, gate errors, and decoherence. Optimizing the implementation of the IQFT is crucial for improving the performance of quantum algorithms that rely on it.

Open QASM reference code: cz q[0], q[1];

CZ Gate(Controlled-Z)

CZ Gate Matrix

The Controlled-Z (CZ) gate is a fundamental two-qubit gate in quantum computing that induces a controlled phase shift between the states |1⟩ and |-1⟩ while leaving other computational basis states unchanged. It’s one of the key building blocks for implementing quantum algorithms and performing quantum computations.

The Controlled-Z (CZ) gate is a fundamental two-qubit quantum gate. It’s a type of controlled-phase gate, meaning it applies a phase shift to the target qubit only when the control qubit is in the |1⟩ state.

How it Works:

Control Qubit: Determines whether the operation is applied.
Target Qubit: The qubit that is affected by the phase shift.

If the control qubit is in the |0⟩ state, nothing happens to the target qubit. If the control qubit is in the |1⟩ state, a phase shift of π radians (or -1) is applied to the target qubit.

Mathematical Representation:

The following unitary matrix can represent the CZ gate:

|00⟩  |01⟩  |10⟩  |11⟩
--------------------------------
|1   0   0   0|
|0   1   0   0|
|0   0   1   0|
|0   0   0  -1|

Circuit Symbol:

The CZ gate is typically represented by the following circuit symbol:

  ┌───┐
q_0:┤ C ├
     │
q_1:┤ Z ├
     └───┘

Key Points:

Phase Flip: The CZ gate essentially flips the phase of the target qubit when the control qubit is in the |1⟩ state.
Quantum Entanglement: It can be used to create entangled states between qubits.
Quantum Algorithms: The CZ gate is a crucial component in many quantum algorithms, including quantum error correction and quantum simulation.

In essence, the CZ gate is a powerful tool for manipulating quantum information and creating complex quantum states.

The CZ gate performs a phase shift of π (or equivalently, a phase factor of -1) on the target qubit if and only if the control qubit is in the state |1⟩. If the control qubit is in the state |0⟩, no phase shift is applied to the target qubit.

The CZ gate is commonly used to create entanglement between qubits, which is essential for various quantum algorithms and protocols, including quantum teleportation, superdense coding, and quantum error correction.

Open QASM reference code: cu1 (pi / 2) q[0], q[1];

CP Gate(Controlled Phase)

A Controlled-Phase (CP) gate is a two-qubit quantum gate that applies a phase shift to the target qubit only when the control qubit is in the |1⟩ state. It’s a fundamental building block in quantum circuits, often used in quantum algorithms like quantum Fourier transform and quantum error correction.

Mathematical Representation: The CP gate can be represented by a 4×4 unitary matrix:

|00⟩  |01⟩  |10⟩  |11⟩
--------------------------------
|1   0   0   0|
|0   1   0   0|
|0   0   1   0|
|0   0   0   e^(iθ)|

Here, θ is the phase shift applied to the target qubit when the control qubit is in the |1⟩ state.

Circuit Symbol: The CP gate is typically represented by the following circuit symbol:

  ┌───┐
q_0:┤ C ├
     │
q_1:┤ P ├
     └───┘

Where:

C: Control qubit
P: Target qubit

Key Points:

Phase Shift: The CP gate introduces a relative phase shift between the |11⟩ and |00⟩ states.
Control Qubit: The control qubit determines whether the phase shift is applied or not.
Target Qubit: The target qubit is the one that undergoes the phase shift.
Applications:
- Quantum Fourier Transform
- Quantum Error Correction
- Quantum Phase Estimation
- Many other quantum algorithms

By carefully controlling the phase shifts applied to qubits, quantum computers can perform complex calculations that are difficult or impossible for classical computers. The CP gate is a crucial tool in achieving this quantum advantage. Its primary function is to introduce a phase shift to the target qubit, conditioned on the state of the control qubit.

Why is Phase Shift Important?

Quantum Interference: Phase shifts are essential for creating quantum interference, a phenomenon that allows quantum computers to perform calculations exponentially faster than classical computers.
Quantum Algorithms: Many quantum algorithms, such as the Quantum Fourier Transform (QFT) and Shor’s algorithm, heavily rely on phase shifts to manipulate quantum states.

Applications of the CP Gate:

Quantum Fourier Transform (QFT):
- The QFT is a fundamental subroutine in many quantum algorithms, including Shor’s algorithm for factoring integers.
- It involves a series of controlled-phase rotations to decompose a quantum state into its frequency components.
Quantum Error Correction:
- Quantum error correction codes use CP gates and other quantum gates to detect and correct errors in quantum information.
- By introducing controlled phase shifts, these codes can identify and mitigate the effects of noise and decoherence.
Quantum Phase Estimation:
- Quantum phase estimation is a technique used to estimate the eigenvalues of a unitary operator.
- It involves a series of controlled-phase rotations to gradually accumulate phase shifts, which can then be measured to determine the eigenvalue.
Quantum Teleportation:
- Quantum teleportation is a process for transmitting quantum information over a distance.
- It involves a series of quantum gates, including the CP gate, to transfer the quantum state of one qubit to another qubit.

Open QASM reference code: cu1 (pi / 2) q[0], q[1];

CY Gate(Controlled Y-Gate)

CY Gate Matrix

The CY gate in quantum computing is a controlled Y gate. It is a two-qubit gate where the Pauli-Y gate is applied to the target qubit only if the control qubit is in the state |1⟩. The CY gate is a specific case of a controlled unitary gate, similar to the more commonly known controlled-X (CNOT) gate, but instead applies the Pauli-Y operation to the target qubit. A 4×4 matrix represents it and is useful in various quantum computing tasks, including state manipulation, entanglement generation, and error correction.

If the control qubit is 0, the target qubit remains unchanged.
If the control qubit is 1, the target qubit is flipped (0 becomes 1, and 1 becomes 0).

Pauli-Y Gate

In quantum computing, a qubit can exist in a superposition of two states, |0⟩ and |1⟩. A two-qubit system can be represented by a four-dimensional state vector:

|ψ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩

where α, β, γ, and δ are complex numbers such that |α|² + |β|² + |γ|² + |δ|² = 1.

The CY Gate Matrix:

The CY gate, denoted by CY, can be represented by a 4×4 unitary matrix:

CY = 
| 1  0  0  0 |
| 0  1  0 -i |
| 0  0  0  0 |
| 0  i  0  0 |

How the CY Gate Operates

When applied to a two-qubit state |ψ⟩, the CY gate transforms it as follows:

CY |ψ⟩ = CY (α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩)

By applying the matrix multiplication, we get:

= α|00⟩ + β|01⟩ - iγ|10⟩ + iδ|11⟩

Interpretation:

If the control qubit is in the |0⟩ state, the target qubit remains unchanged.
If the control qubit is in the |1⟩ state, the target qubit undergoes a phase shift of -π/2 (for the |10⟩ state) or π/2 (for the |11⟩ state).

Applications of CY Gate

Entanglement Generation: The CY gate can be used to create entangled states, particularly when combined with other gates like Hadamard and CNOT gates.
Quantum Algorithms: In certain quantum algorithms, the CY gate may be used to introduce specific phase shifts or as part of more complex controlled operations.
Quantum Fourier Transform: Used to decompose functions into a basis of periodic functions.
Shor’s algorithm is used to factor large numbers efficiently.
Grover’s algorithm: Used for searching unsorted databases efficiently.
Quantum Error Correction: CY gates can be used in quantum error correction codes and protocols that require specific phase corrections depending on the state of qubits.

Open QASM reference code: cy q[1], q[2];

C3X Gate(Triple controlled X gate)

C3X Gate Matrix

C3X (Controlled-Controlled-X) gate is a fundamental three-qubit quantum gate. It’s essentially a Toffoli gate with an additional control qubit. Its primary function is to conditionally flip the target qubits based on the states of the two control qubits.

Operation Breakdown:

Control Qubits: The gate will have an effect if both control qubits are in the |1> state.
Target Qubit: If the condition above is met, the target qubit will be flipped (|0> becomes |1> and vice versa). If either control qubit is in the |0> state, the target qubit remains unchanged.

Matrix Representation:

An 8×8 unitary matrix can represent the C3X gate. Each row and column corresponds to a possible state of the three qubits. The diagonal elements represent the probability amplitudes of the corresponding states after applying the gate.

Circuit Diagram:

In quantum circuit diagrams, the C3X gate is typically depicted as a controlled-X gate (CNOT) with an additional control qubit.

   control1  control2  target
     |       |       |
     ------X------X

Applications:

Quantum Error Correction: The C3X gate is essential for building quantum error-correcting codes, which protect quantum information from noise and decoherence.
Quantum Algorithms: It’s used as a building block in various quantum algorithms, such as Shor’s algorithm for factoring large numbers and Grover’s algorithm for quantum search.
Quantum Computing Architecture: The C3X gate, along with other basic gates, forms the foundation of many quantum computing architectures.

Key Points to Remember:

The C3X gate is a three-qubit gate.
It requires both control qubits to be in the |1> state for the target qubit to be flipped.
It’s a crucial component in quantum error correction and various quantum algorithms.

C4X Gate(4-qubit controlled X gate)

The C4X gate, also known as the Controlled-Controlled-Controlled-X gate, is a four-qubit quantum gate in quantum computing that performs the following operation:

C4X(control1, control2, control3, target):

If all three control qubits are 1, the target qubit is flipped (0 becomes 1, and 1 becomes 0). If any of the control qubits are 0, the target qubit remains unchanged.

The C4X gate is a quantum logic gate used in quantum computing, where C4X stands for “Controlled-4 X.” It’s a generalization of the CNOT (Controlled-NOT) gate to more than one control qubit.

Understanding the C4X Gate

Basic Concept:
- A CNOT gate flips the state of a target qubit if the control qubit is in the state |1⟩.
- A C4X gate extends this concept to four control qubits. The target qubit is flipped if all four control qubits are in the state |1⟩.
Mathematical Representation:
- The C4X gate can be represented using a matrix, though it becomes a very large matrix for 5 qubits (4 control + 1 target). The basic operation can be expressed as:
  - If all four control qubits are in the state |1⟩, the target qubit undergoes an X operation (NOT operation).
  - If any of the control qubits are in the state |0⟩, the target qubit remains unchanged.
Circuit Symbol:
- The C4X gate is typically represented in a quantum circuit diagram with four control lines connected to a single target line. The controls are indicated by dots (•), and the target is marked with an X.
Applications:
- C4X gates are used in more complex quantum algorithms where multi-qubit control operations are necessary. For example, in error correction, quantum cryptography, or certain types of quantum search algorithms.
- These gates can be part of a larger decomposition in quantum circuits that require precise control based on multiple qubits.
Decomposition:
- In practical quantum circuits, implementing a C4X gate directly can be challenging due to hardware limitations. Therefore, it is often decomposed into a sequence of simpler gates, such as CNOT gates, Toffoli gates (CCX), and other elementary operations.

C5X Gate(5-qubit controlled X gate)

The C5x gate refers to a controlled gate operation in quantum computing with five control qubits. In general, a CkX gate is a controlled gate where “k” refers to the number of control qubits, and “X” refers to the Pauli-X (also known as the NOT gate) operation on a target qubit. The operation of the C5X gate can be understood as follows:

1. Control Qubits (5 qubits):

These five qubits determine whether the Pauli-X operation will be applied to the target qubit.
The C5X gate will only flip the state of the target qubit if all five control qubits are in the ∣1⟩ state (i.e., all are in the state representing the binary value 1).

2. Target Qubit:

This is the qubit that will be flipped if the condition on the control qubits is satisfied.

4. Quantum Circuit Representation:

In a quantum circuit, the C5X gate is often represented by a series of control qubits connected to a target qubit with a line ending in an X symbol.
This gate can be decomposed into a series of CNOT (CX) gates and other quantum gates to be implemented on quantum hardware since the direct implementation of a C5X gate may not be straightforward on most quantum computers.

5. Applications:

Such multi-controlled gates are used in complex quantum algorithms, such as Grover’s search algorithm, where multiple conditions must be checked simultaneously before performing a specific operation.
They are also essential in error correction codes and other quantum operations that require conditional logic based on multiple qubits.

Unitary Gate(U3 Gate)

U3 Gate Matrix

The U3 gate is one of the most general single-qubit quantum gates used in quantum computing. It is a universal gate, meaning that any single-qubit operation can be expressed as a U3 gate with appropriate parameters.

Matrix representation:

A 2×2 unitary matrix can represent the U3 gate:

|cos(θ/2)   -i*sin(θ/2)*e^(-i*φ) |
|i*sin(θ/2)*e^(i*φ)   cos(θ/2)   |

where:

θ (theta) is a real number between 0 and π (0 and 180 degrees).
φ (phi) is a real number between 0 and 2π (0 and 360 degrees).
λ (lambda) is a real number between 0 and 2π (0 and 360 degrees).

Circuit diagram:

The U3 gate is often represented in quantum circuit diagrams as follows:

|U3(θ, φ, λ)

Applications:

The U3 gate is used in various quantum algorithms and circuits, including:

Quantum state preparation: Used to create specific quantum states.
Quantum gates: Can be used to implement other quantum gates, such as the Pauli gates (X, Y, Z) and the Hadamard gate.
Quantum algorithms: Used as a building block in many quantum algorithms, such as Shor’s algorithm for factoring large numbers and Grover’s algorithm for searching unsorted databases.

Additional details:

The U3 gate can be decomposed into a sequence of other gates, such as rotations around the X, Y, and Z axes.
The U3 gate is often used in conjunction with other gates to create more complex quantum operations.
The parameters θ, φ, and λ of the U3 gate can be chosen to represent a wide variety of quantum operations.

Gate Pallet

Hadamard Gate(H gate)

Y gate

Z gate

T gate

S gate

P Gate

$T^{†}$ gate

$S^{†}$ gate

RX gate

RY gate

RZ gate

NOT Gate(Pauli-X)

CNOT Gate(CX)

Toffoli gate(CCX)

Identity gate

SWAP gate

Measurement Gate

Reset Gate

Barrier operation

Quantum Fourier Transform(QFT)

Inverse Quantum Fourier Transform(iQFT)

CZ Gate(Controlled-Z)

CP Gate(Controlled Phase)

CY Gate(Controlled Y-Gate)

C3X Gate(Triple controlled X gate)

C4X Gate(4-qubit controlled X gate)

C5X Gate(5-qubit controlled X gate)

Unitary Gate(U3 Gate)

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Gate Pallet

Hadamard Gate(H gate)

Y gate

Z gate

T gate

S gate

P Gate

T† gate

S† gate

RX gate

RY gate

RZ gate

NOT Gate(Pauli-X)

CNOT Gate(CX)

Toffoli gate(CCX)

Identity gate

SWAP gate

Measurement Gate

Reset Gate

Barrier operation

Quantum Fourier Transform(QFT)

Inverse Quantum Fourier Transform(iQFT)

CZ Gate(Controlled-Z)

CP Gate(Controlled Phase)

CY Gate(Controlled Y-Gate)

C3X Gate(Triple controlled X gate)

C4X Gate(4-qubit controlled X gate)

C5X Gate(5-qubit controlled X gate)

Unitary Gate(U3 Gate)

$T^{†}$ gate

$S^{†}$ gate