Quantum logic gate

In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

Unlike many classical logic gates, quantum logic gates are reversible. However, it is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancilla bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits.

Quantum gates are unitary operators, and are described as unitary matrices relative to some basis. Usually we use the computational basis, which unless we compare it with something, just means that for a d-level quantum system (such as a qubit, a quantum register, or qutrits and qudits[1]:22–23) we have labeled the orthogonal basis vectors , or use binary notation.

Common quantum logic gates by name (including abbreviation), circuit form(s) and the corresponding unitary matrices.

HistoryEdit

The current notation for quantum gates was developed by many of the founding fathers of quantum information science including Adriano Barenco, Charles Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter,[2] building on notation introduced by Richard Feynman.[3]

RepresentationEdit

 
Single qubit states that are pure (i.e. lacks phase shifts along the basis axes, and is not entangled) can be represented using the Bloch sphere, writing it as  
Rotation about the x, y, z-axes of the Bloch sphere is represented by the rotation operator gates.

Quantum logic gates are represented by unitary matrices. A gate which acts on   qubits is represented by a   unitary matrix, and the set of all such gates with the group operation of matrix multiplication[a] is the symmetry group U(2n). The quantum states that the gates act upon are unit vectors in   complex dimensions, where the norm is the modulus squared. The basis vectors are the possible outcomes if measured, and a quantum state is a linear combination of these outcomes. The most common quantum gates operate on spaces of one or two qubits, just like the common classical logic gates operate on one or two bits.

Quantum states are typically represented by "kets", from a mathematical notation known as bra-ket.

The vector representation of a single qubit is:

 

Here,   and   are the complex probability amplitudes of the qubit. These values determine the probability of measuring a 0 or a 1, when measuring the state of the qubit. See measurement below for details.

The value zero is represented by the ket  , and the value one is represented by the ket  .

The tensor product (or Kronecker product) is used to combine quantum states. The combined state of two qubits is the tensor product of the two qubits. The tensor product is denoted by the symbol  .

The vector representation of two qubits is:

 

The action of the gate on a specific quantum state is found by multiplying the vector   which represents the state, by the matrix   representing the gate. The result is a new quantum state  :

 

Notable examplesEdit

There exist an infinite number of gates. Some of them have been named by various authors,[1][4][5][6][7][8] and below follow some of those most often used in the literature.

Identity gateEdit

The identity gate is the identity matrix, usually written as I, and is defined for a single qubit as

 

where I is basis independent and does not modify the quantum state. The identity gate is most useful when describing mathematically the result of various gate operations or when discussing multi-qubit circuits.

Pauli gates (X,Y,Z)Edit

The Pauli gates   are the three Pauli matrices   and act on a single qubit. The Pauli X,Y and Z equate, respectively, to a rotation around the x, y and z axes of the Bloch sphere by   radians.

The Pauli-X gate is the quantum equivalent of the NOT gate for classical computers with respect to the standard basis  ,  , which distinguishes the z-axis on the Bloch sphere. It is sometimes called a bit-flip as it maps   to   and   to  . Similarly, the Pauli-Y maps   to   and   to  . Pauli Z leaves the basis state   unchanged and maps   to  . Due to this nature, it is sometimes called phase-flip.

These matrices are usually represented as

 
 
 

The Pauli matrices are involutory, meaning that the square of a Pauli matrix is the identity matrix.

 

The Pauli matrices also anti-commute, for example   .

Square root of NOT gate (NOT)Edit

 
Quantum circuit diagram of a square-root-of-NOT gate

The square root of NOT gate (or square root of Pauli-X,  ) acts on a single qubit. It maps the basis state   to   and   to  . In matrix form it is given by

 ,

such that

 

This operation represents a rotation of π/2 about x-axis at the Bloch sphere.

Controlled gatesEdit

 
Circuit diagram of controlled Pauli gates (from left to right): CNOT (or controlled-X), controlled-Y and controlled-Z.

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation.[2] For example, the controlled NOT gate (or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is  , and otherwise leaves it unchanged. With respect to the basis  ,  ,  ,  , it is represented by the matrix:

 

The CNOT (or controlled Pauli-X) gate can be described as the gate that maps the basis states  , where   is XOR.

More generally if U is a gate that operates on single qubits with matrix representation

 

then the controlled-U gate is a gate that operates on two qubits in such a way that the first qubit serves as a control. It maps the basis states as follows.

 
Circuit representation of controlled-U gate
 
 
 
 

The matrix representing the controlled U is

 

When U is one of the Pauli operators, X,Y, Z, the respective terms "controlled-X", "controlled-Y", or "controlled-Z" are sometimes used.[4]:177–185 Sometimes this is shortened to just CX, CY and CZ.

Phase shift gatesEdit

The phase shift is a family of single-qubit gates that map the basis states   and  . The probability of measuring a   or   is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to conditionally tracing a horizontal circle (a line of latitude) on the Bloch sphere by   radians.[b] The phase shift gate is represented by the matrix:

 

where   is the phase shift with the period . Some common examples are the T gate where  , the phase gate (written S, though S is sometimes used for SWAP gates) where   and the Pauli-Z gate where  .

The phase shift gates are related to each other as follows:

 
 
 
  for all real   except 0[c]

The argument to the phase shift gate is in U(1), and the gate performs a phase rotation in U(1) along the specified basis axis (e.g.   rotates the phase about the  -axis). U(1) is a subgroup of U(n) and contains the phase of the quantum system. Extending   to a rotation about a generic phase of both axes of a 2-level quantum system (a qubit) can be done with a series circuit:  . When   this gate is the rotation operator   gate.[d]

Introducing the global phase gate  [e] we also have the identity  .[2]:11[1]:77–83

Arbitrary single-qubit phase shift gates   are natively available for transmon quantum processors through timing of microwave control pulses.[9]

Controlled phase shiftEdit

The 2-qubit controlled phase shift gate is:

 

With respect to the computational basis, it shifts the phase with   only if it acts on the state  :

 

The CZ gate is the special case where  .

Rotation operator gatesEdit

The rotation operator gates   and   are the analog rotation matrices in three Cartesian axes of SO(3), the axes on the Bloch sphere projection:

 
 
 

As Pauli matrices are related to the generator of rotations, these rotation operators can be written as matrix exponentials with Pauli matrices in the argument. Any   unitary matrix in SU(2) can be written as a product (i.e. series circuit) of three rotation gates or less. Note that for two-level systems such as qubits and spinors, these rotations have a period of . A rotation of (360 degrees) returns the same statevector with a different phase.[10]

We also have   and   for all  

The rotation matrices are related to the Pauli matrices in the following way :  

Hadamard gateEdit

The Hadamard gate (French: [adamaʁ]) acts on a single qubit. It maps the basis state   to   and   to  , which means that a measurement will have equal probabilities to result in 1 or 0 (i.e. creates a superposition). It represents a rotation of   about the axis   at the Bloch sphere. It is represented by the Hadamard matrix:

 
Circuit representation of Hadamard gate
 

H is an involutory matrix. Using rotation operators, we have the identities:   and   Controlled-H gate can also be defined as explained in the section of controlled gates.

The Hadamard gate can be thought as a unitary transformation that maps qubit operations in z-axis to the x-axis and viceversa. For example,  ,  , and  .

Swap gateEdit

 
Circuit representation of SWAP gate

The swap gate swaps two qubits. With respect to the basis  ,  ,  ,  , it is represented by the matrix:

 

Square root of swap gateEdit

 
Circuit representation of SWAP gate

The SWAP gate performs half-way of a two-qubit swap. It is universal such that any many-qubit gate can be constructed from only SWAP and single qubit gates. The SWAP gate is not, however maximally entangling; more than one application of it is required to produce a Bell state from product states. With respect to the basis  ,  ,  ,  , it is represented by the matrix:

 

This gate arises naturally in systems that exploit exchange interaction.[11][12]

Toffoli (CCNOT) gateEdit

 
Circuit representation of Toffoli gate

The Toffoli gate, named after Tommaso Toffoli; also called CCNOT gate or Deutsch gate  ; is a 3-bit gate, which is universal for classical computation but not for quantum computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If we limit ourselves to only accepting input qubits that are   and  , then if the first two bits are in the state   it applies a Pauli-X (or NOT) on the third bit, else it does nothing. It is an example of a controlled gate. Since it is the quantum analog of a classical gate, it is completely specified by its truth table. The Toffoli gate is universal when combined with the single qubit Hadamard gate.[13]

Truth table Matrix form
INPUT OUTPUT
0 0 0 0 0 0
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 0 1
1 1 0 1 1 1
1 1 1 1 1 0

 

The quantum Toffoli gate is also related to the classical AND operation as it can also described as the gate with the mapping   for states in the computational basis.

Fredkin (CSWAP) gateEdit

 
Circuit representation of Fredkin gate

The Fredkin gate (also CSWAP or CS gate), named after Edward Fredkin, is a 3-bit gate that performs a controlled swap. It is universal for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the billiard ball model means the same number of balls are output as input.

Truth table Matrix form
INPUT OUTPUT
C I1 I2 C O1 O2
 0   0   0   0   0   0 
0 0 1 0 0 1
0 1 0 0 1 0
0 1 1 0 1 1
1 0 0 1 0 0
1 0 1 1 1 0
1 1 0 1 0 1
1 1 1 1 1 1

 

Ising coupling gatesEdit

The Ising coupling gates Rxx, Ryy and Rzz are 2-qubit gates that are implemented natively in some trapped-ion quantum computers.[14][15] These gates are defined as

 
 

 [16]

Imaginary swap (iSWAP)Edit

For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap[17] or iSWAP gate defined as[18][19]

 

where its squared root version is given by

 

Deutsch gateEdit

The Deutsch (or  ) gate, named after physicist David Deutsch is a three-qubit gate. It is defined as

 

Unfortunately, a working Deutsch gate has remained out of reach, due to lack of a protocol. There are some proposals to realize a Deutsch gate with dipole-dipole interaction in neutral atoms.[20]

Universal quantum gatesEdit

 
Both CNOT and   are universal two-qubit gates and can be transformed into each other.

A set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Technically, this is impossible with anything less than an uncountable set of gates since the number of possible quantum gates is uncountable, whereas the number of finite sequences from a finite set is countable. To solve this problem, we only require that any quantum operation can be approximated by a sequence of gates from this finite set. Moreover, for unitaries on a constant number of qubits, the Solovay–Kitaev theorem guarantees that this can be done efficiently.

The rotation operators Rx(θ), Ry(θ), Rz(θ), the phase shift gate P(φ) and CNOT form a widely used universal set of quantum gates.[12]

A common universal gate set is the Clifford + T gate set, which is composed of the CNOT, H, S and T gates. The Clifford set alone is not universal and can be efficiently simulated classically by the Gottesman-Knill theorem.

The Toffoli gate forms a set of universal gates for reversible logic circuits. A two-gate set of universal quantum gates containing a Toffoli gate can be constructed by adding the Hadamard gate to the set.[21]

A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate  .[22]

A universal logic gate for reversible classical computing, the Toffoli gate, is reducible to the Deutsch gate,  , thus showing that all reversible classical logic operations can be performed on a universal quantum computer.

There also exists a single two-qubit gate sufficient for universality, given it can be applied to any pairs of qubits   on a circuit of width  .[23]

Circuit compositionEdit

Serially wired gatesEdit

 
Two gates Y and X in series. The order in which they appear on the wire is reversed when multiplying them together.

Assume that we have two gates A and B, that both act on   qubits. When B is put after A in a series circuit, then the effect of the two gates can be described as a single gate C.

 

Where   is matrix multiplication. The resulting gate C will have the same dimensions as A and B. The order in which the gates would appear in a circuit diagram is reversed when multiplying them together.[4]:17–18,22–23,62–64[5]:147–169

For example, putting the Pauli X gate after the Pauli Y gate, both of which act on a single qubit, can be described as a single combined gate C:

 

The product symbol ( ) is often omitted.

Exponents of quantum gatesEdit

All real exponents of unitary matrices are also unitary matrices, and all quantum gates are unitary matrices.

Positive integer exponents are equivalent to sequences of serially wired gates (e.g.  ), and the real exponents is a generalization of the series circuit. For example,   and   are both valid quantum gates.

  for any unitary matrix  . The identity matrix ( ) acts like a NOP and can be represented as bare wire in quantum circuits, or not shown at all.

All gates are unitary matrices, so that   and  , where   is the conjugate transpose. This means that negative exponents of gates are unitary inverses of their positively exponentiated counterparts:  . For example, some negative exponents of the phase shift gates are   and  .

Parallel gatesEdit

 
Two gates   and   in parallel is equivalent to the gate  

The tensor product (or Kronecker product) of two quantum gates is the gate that is equal to the two gates in parallel.[4]:71–75[5]:148

If we, as in the picture, combine the Pauli-Y gate with the Pauli-X gate in parallel, then this can be written as:

 

Both the Pauli-X and the Pauli-Y gate act on a single qubit. The resulting gate   act on two qubits.

Hadamard transformEdit

The gate   is the Hadamard gate ( ) applied in parallel on 2 qubits. It can be written as:

 

This "two-qubit parallel Hadamard gate" will when applied to, for example, the two-qubit zero-vector ( ), create a quantum state that have equal probability of being observed in any of its four possible outcomes;  ,  ,  , and  . We can write this operation as:

 
 
Example: The Hadamard transform on a 3-qubit register  . Qubit 0 is least significant.

Here the amplitude for each measurable state is 12. The probability to observe any state is the square of the absolute value of the measurable states amplitude, which in the above example means that there is one in four that we observe any one of the individual four cases. See measurement for details.

  performs the Hadamard transform on two qubits. Similarly the gate   performs a Hadamard transform on a register of   qubits.

When applied to a register of   qubits all initialized to  , the Hadamard transform puts the quantum register into a superposition with equal probability of being measured in any of its   possible states:

 

This state is a uniform superposition and it is generated as the first step in some search algorithms, for example in amplitude amplification and phase estimation.

Measuring this state results in a random number between   and  .[f] How random the number is depends on the fidelity of the logic gates. If not measured, it is a quantum state with equal probability amplitude   for each of its possible states.

The Hadamard transform acts on a register   with   qubits such that   as follows:

 

Application on entangled statesEdit

If two or more qubits are viewed as a single quantum state, this combined state is equal to the tensor product of the constituent qubits. Any state that can be written as a tensor product from the constituent subsystems are called separable states (pure states). On the other hand, an entangled state is any state that cannot be tensor-factorized, or in other words: An entangled state can not be written as a tensor product of its constituent qubits states. Special care must be taken when applying gates to constituent qubits that make up entangled states.

If we have a set of N qubits that are entangled and wish to apply a quantum gate on M < N qubits in the set, we will have to extend the gate to take N qubits. This application can be done by combining the gate with an identity matrix such that their tensor product becomes a gate that act on N qubits. The identity matrix ( ) is a representation of the gate that maps every state to itself (i.e., does nothing at all). In a circuit diagram the identity gate or matrix will often appear as just a bare wire.

 
The example given in the text. The Hadamard gate   only act on 1 qubit, but   is an entangled quantum state that spans 2 qubits. In our example,  

For example, the Hadamard gate ( ) acts on a single qubit, but if we for example feed it the first of the two qubits that constitute the entangled Bell state  , we cannot write that operation easily. We need to extend the Hadamard gate   with the identity gate   so that we can act on quantum states that span two qubits:

 

The gate   can now be applied to any two-qubit state, entangled or otherwise. The gate   will leave the second qubit untouched and apply the Hadamard transform to the first qubit. If applied to the Bell state in our example, we may write that as:

 

Computational complexity and the tensor productEdit

The time complexity for multiplying two  -matrices is at least  ,[24] if using a classical machine. Because the size of a gate that operates on   qubits is   it means that the time for simulating a step in a quantum circuit (by means of multiplying the gates) that operates on generic entangled states is  . For this reason it is believed to be intractable to simulate large entangled quantum systems using classical computers. Subsets of the gates, such as the Clifford gates, can however be efficiently simulated on classical computers.

The state vector of a quantum register with   qubits is   complex entries. Storing the probability amplitudes as a list of floating point values is not tractable for large  .

Unitary inversion of gatesEdit

 
Example: The unitary inverse of the Hadamard-CNOT product. The three gates  ,   and   are their own unitary inverses.

Because all quantum logical gates are reversible, any composition of multiple gates is also reversible. All products and tensor products (i.e. series and parallel combinations) of unitary matrices are also unitary matrices. This means that it is possible to construct an inverse of all algorithms and functions, as long as they contain only gates.

Initialization, measurement, I/O and spontaneous decoherence are side effects in quantum computers. Gates however are purely functional and bijective.

If   is a unitary matrix, then   and  . The dagger ( ) denotes the complex conjugate of the transpose. It is also called the Hermitian adjoint.

If a function   is a product of   gates,  , the unitary inverse of the function   can be constructed:

Because   we have, after repeated application on itself

 

Similarly if the function   consists of two gates   and   in parallel, then   and  .

Gates that are their own unitary inverses are called Hermitian or self-adjoint operators. Some elementary gates such as the Hadamard (H) and the Pauli gates (I, X, Y, Z) are Hermitian operators, while others like the phase shift (S, T, P, CPHASE) gates generally are not. Gates that are not Hermitian are sometimes called skew-Hermitian, or adjoint operators.

For example, an algorithm for addition can be used for subtraction, if it is being "run in reverse", as its unitary inverse. The inverse quantum fourier transform is the unitary inverse. Unitary inverses can also be used for uncomputation. Programming languages for quantum computers, such as Microsoft's Q#[25] and Bernhard Ömer's QCL[26]:61, contain function inversion as programming concepts.

MeasurementEdit

 
Circuit representation of measurement. The two lines on the right hand side represent a classical bit, and the single line on the left hand side represents a qubit.

Measurement (sometimes called observation) is irreversible and therefore not a quantum gate, because it assigns the observed quantum state to a single value. Measurement takes a quantum state and projects it to one of the basis vectors, with a likelihood equal to the square of the vector's depth (the norm is the modulus squared) along that basis vector.[1]:15–17[27][28][29] This is known as the Born rule and appears[f] as a stochastic non-reversible operation as it probabilistically sets the quantum state equal to the basis vector that represents the measured state (the state "collapses" to a definite single value). Why and how, or even if[30][31] the quantum state collapses at measurement, is called the measurement problem.

The probability of measuring a value with probability amplitude   is  , where   is the modulus.

Measuring a single qubit, whose quantum state is represented by the vector  , will result in   with probability  , and in   with probability  .

For example, measuring a qubit with the quantum state   will yield with equal probability either   or  .

 
For a single qubit, we have a unit sphere in   with the quantum state   such that  . The state can be re-written as  , or   and  .
Note:   is the probability of measuring   and   is the probability of measuring  .

A quantum state   that spans n qubits can be written as a vector in   complex dimensions:  . This is because the tensor product of n qubits is a vector in   dimensions. This way, a register of n qubits can be measured to   distinct states, similar to how a register of n classical bits can hold   distinct states. Unlike with the bits of classical computers, quantum states can have non-zero probability amplitudes in multiple measurable values simultaneously. This is called superposition.

The sum of all probabilities for all outcomes must always be equal to 1.[g] Another way to say this is that the Pythagorean theorem generalized to   has that all quantum states   with n qubits must satisfy  , where   is the probability amplitude for measurable state  . A geometric interpretation of this is that the possible value-space of a quantum state   with n qubits is the surface of a unit sphere in   and that the unitary transforms (i.e. quantum logic gates) applied to it are rotations on the sphere. The rotations that the gates perform is in the symmetry group U(2n). Measurement is then a probabilistic projection of the points at the surface of this complex sphere onto the basis vectors that span the space (and labels the outcomes).

In many cases the space is represented as a Hilbert space   rather than some specific  -dimensional complex space. The number of dimensions (defined by the basis vectors, and thus also the possible outcomes from measurement) is then often implied by the operands, for example as the required state space for solving a problem. In Grover's algorithm, Lov named this generic basis vector set "the database".

The selection of basis vectors against to measure a quantum state will influence the outcome of the measurement.[1]:30–35[4]:22,84–85,185–188[32] See Von Neumann entropy for details. In this article, we always use the computational basis, which means that we have labeled the   basis vectors of an n-qubit register  , or use the binary representation  .

In the quantum computing domain, it is generally assumed that the basis vectors constitute an orthonormal basis.

An example of usage of an alternative measurement basis is in the BB84 cipher.

The effect of measurement on entangled statesEdit

 
The Hadamard-CNOT gate, which when given the input   produces a Bell state.

If two quantum states (i.e. qubits, or registers) are entangled (meaning that their combined state cannot be expressed as a tensor product), measurement of one register affects or reveals the state of the other register by partially or entirely collapsing its state too. This effect can be used for computation, and is used in many algorithms.

The Hadamard-CNOT combination acts on the zero-state as follows:

 
 
The Bell state in the text is   where   and  . Therefore it can be described by the complex plane spanned by the basis vectors   and  , as in the picture. The unit sphere (in  ) that represent the possible value-space of the 2-qubit system intersects the plane and   lies on the unit spheres surface. Because  , there is equal probability of measuring this state to   or  , and zero probability of measuring it to   or  .

This resulting state is the Bell state  . It cannot be described as a tensor product of two qubits. There is no solution for

 

because for example w needs to be both non-zero and zero in the case of xw and yw.

The quantum state spans the two qubits. This is called entanglement. Measuring one of the two qubits that make up this Bell state will result in that the other qubit logically must have the same value, both must be the same: Either it will be found in the state  , or in the state  . If we measure one of the qubits to be for example  , then the other qubit must also be  , because their combined state became  . Measurement of one of the qubits collapses the entire quantum state, that span the two qubits.

The GHZ state is a similar entangled quantum state that spans three or more qubits.

This type of value-assignment occurs instantaneously over any distance and this has as of 2018 been experimentally verified by QUESS for distances of up to 1200 kilometers.[33][34][35] That the phenomena appears to happen instantaneously as opposed to the time it would take to traverse the distance separating the qubits at the speed of light is called the EPR paradox, and it is an open question in physics how to resolve this. Originally it was solved by giving up the assumption of local realism, but other interpretations have also emerged. For more information see the Bell test experiments. The no-communication theorem proves that this phenomena cannot be used for faster-than-light communication of classical information.

Measurement on registers with pairwise entangled qubitsEdit

 
The effect of a unitary transform F on a register A that is in a superposition of   states and pairwise entangled with the register B. Here, n is 3 (each register has 3 qubits).

Take a register A with n qubits all initialized to  , and feed it through a parallel Hadamard gate  . Register A will then enter the state   that have equal probability of when measured to be in any of its   possible states;   to  . Take a second register B, also with n qubits initialized to   and pairwise CNOT its qubits with the qubits in register A, such that for each p the qubits   and   forms the state  .

If we now measure the qubits in register A, then register B will be found to contain the same value as A. If we however instead apply a quantum logic gate F on A and then measure, then  , where   is the unitary inverse of F.

Because of how unitary inverses of gates act,  . For example, say  , then  .

The equality will hold no matter in which order measurement is performed (on the registers A or B), assuming that F has run to completion. Measurement can even be randomly and concurrently interleaved qubit by qubit, since the measurements assignment of one qubit will limit the possible value-space from the other entangled qubits.

Even though the equalities holds, the probabilities for measuring the possible outcomes may change as a result of applying F, as may be the intent in a quantum search algorithm.

This effect of value-sharing via entanglement is used in Shor's algorithm, phase estimation and in quantum counting. Using the Fourier transform to amplify the probability amplitudes of the solution states for some problem is a generic method known as "Fourier fishing".

Logic function synthesisEdit

Function and routines that only use gates can themselves be described as matrices, just like the smaller gates. The matrix that represents a quantum function that act on   qubits have the size  . For example, a function that act on a "qubyte" (a register of 8 qubits) would be described as a matrix with   elements.

Unitary transformations that are not in the set of gates natively available at the quantum computer (the primitive gates) can be synthesised, or approximated, by combining the available primitive gates in a circuit. One way to do this is to factorize the matrix that encodes the unitary transformation into a product of tensor products (i.e. series and parallel circuits) of the available primitive gates. The group U(2q) is the symmetry group for the gates that act on   qubits.[2] Factorization is then the problem of finding a path in U(2q) from the generating set of primitive gates. The Solovay–Kitaev theorem shows that given a sufficient set of primitive gates, there exist an efficient approximate for any gate. For the general case with large number of qubits this direct approach to circuit synthesis is intractable.[36][37]

Because the gates unitary nature, all functions must be reversible and always be bijective mappings of input to output. There must always exist a function   such that  . Functions that are not invertible can be made invertible by adding ancilla qubits to the input or the output, or both. After the function has run to completion, the ancilla qubits can then either be uncomputed or left untouched. Measuring or otherwise collapsing the quantum state of an ancilla qubit (e.g. by re-initializing the value of it, or by its spontaneous decoherence) that have not been uncomputed may result in errors,[38][39] as their state may be entangled with the qubits that are still being used in computations.

Logically irreversible operations, for example addition modulo   of two  -qubit registers a and b,  , can be made logically reversible by adding information to the output, so that the input can be computed from the output (i.e. there exist a function  ). In our example, this can be done by passing on one of the input registers to the output:  . The output can then be used to compute the input (i.e. given the output   and  , we can easily find the input;   is given and  ) and the function is made bijective.

All boolean algebraic expressions can be encoded as unitary transforms (quantum logic gates), for example by using combinations of the Pauli-X, CNOT and Toffoli gates. These gates are functionally complete in the boolean logic domain.

There are many unitary transforms available in the libraries of Q#, QCL, Qiskit, and other quantum programming languages. It also appears in the literature.[40][41]

For example,  , where   is the number of qubits that constitutes the register  , is implemented as the following in QCL:[42][26]

 
The generated circuit, when  .
The symbols  ,   and   denotes XOR, AND and NOT respectively, and comes from the boolean representation of Pauli-X with zero or more control qubits when applied to states that are in the computational basis.
cond qufunct inc(qureg x) { // increment register
  int i;
  for i = #x-1 to 0 step -1 {
    CNot(x[i], x[0::i]);     // apply controlled-not from
  }                          // MSB to LSB
}

In QCL, decrement is done by "undoing" increment. The undo operator ! is used to instead run the unitary inverse of the function. !inc(x) is the inverse of inc(x) and instead performs the operation  .

In the model of computation used in this article (the quantum circuit model), a classic computer generates the gate composition for the quantum computer, and the quantum computer acts as a coprocessor that receives instructions from the classical computer about which primitive gates to apply to which qubits.[26]:36–43[43] Measurement of quantum registers results in binary values that the classical computer can use in its computations. Quantum algorithms often contain both a classical and a quantum part. Unmeasured I/O (sending qubits to remote computers without collapsing their quantum states) can be used to create networks of quantum computers. Entanglement swapping can then be used to realize distributed algorithms with quantum computers that are not directly connected. Examples of distributed algorithms that only require the use of a handful of quantum logic gates is superdense coding, the quantum Byzantine agreement and the BB84 cipherkey exchange protocol.

See alsoEdit

NotesEdit

  1. ^ Matrix multiplication of quantum gates is defined as series circuits.
  2. ^ It can also be thought of as cutting the Bloch sphere in the X/Y-plane (the equator), and then rotating only one of the two hemispheres of the Bloch sphere (e.g.   rotates the  -hemisphere), leaving the other hemisphere unchanged.
  3. ^   where   is the conjugate transpose (or Hermitian adjoint)
  4. ^   when  , where   is the conjugate transpose (or Hermitian adjoint).
  5. ^ Also:  
  6. ^ a b If this actually is a stochastic effect depends on which interpretation of quantum mechanics that is correct (and if any interpretation can be correct). For example, De Broglie–Bohm theory and the many-worlds interpretation asserts determinism. (In the many-worlds interpretation, a quantum computer is a machine that runs programs (quantum circuits) that selects a reality where the probability of it having the solution states of a problem is large. That is, the machine more often than not exist in a reality where it gives the correct answer. This interpretation does however not change the mechanics by which the machine operates.)
  7. ^ See Probability axioms § Second axiom

ReferencesEdit

  1. ^ a b c d e Colin P. Williams (2011). Explorations in Quantum Computing. Springer. ISBN 978-1-84628-887-6.
  2. ^ a b c d Barenco, Adriano; Bennett, Charles H.; Cleve, Richard; DiVincenzo, David P.; Margolus, Norman; Shor, Peter; Sleator, Tycho; Smolin, John A.; Weinfurter, Harald (1995-11-01). "Elementary gates for quantum computation". Physical Review A. American Physical Society (APS). 52 (5): 3457–3467. arXiv:quant-ph/9503016. doi:10.1103/physreva.52.3457. ISSN 1050-2947.
  3. ^ Feynman, Richard P. (1986). "Quantum mechanical computers". Foundations of Physics. Springer Science and Business Media LLC. 16 (6): 507–531. doi:10.1007/bf01886518. ISSN 0015-9018.
  4. ^ a b c d e Nielsen, Michael A.; Chuang, Isaac (2010). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 978-1-10700-217-3. OCLC 43641333.
  5. ^ a b c Yanofsky, Noson S.; Mannucci, Mirco (2013). Quantum computing for computer scientists. Cambridge University Press. ISBN 978-0-521-87996-5.
  6. ^ "Circuit Library". IBM (Qiskit).
  7. ^ "Quantum inspire Knowledge base". QuTech.
  8. ^ "Azure Quantum Documentation (preview)". Microsoft (Q#).
  9. ^ Dibyendu Chatterjee, Arijit Roy (2015). "A transmon-based quantum half-adder scheme". Progress of Theoretical and Experimental Physics. 2015 (9): 7–8. Bibcode:2015PTEP.2015i3A02C. doi:10.1093/ptep/ptv122.
  10. ^ Griffiths, D.J. (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. pp. 127–128. ISBN 978-3-527-40601-2.
  11. ^ Nemirovsky, Jonathan; Sagi, Yoav (2021), "Fast universal two-qubit gate for neutral fermionic atoms in optical tweezers", Physical Review Research, 3 (1): 013113, arXiv:2008.09819, Bibcode:2021PhRvR...3a3113N, doi:10.1103/PhysRevResearch.3.013113
  12. ^ a b Williams, Colin P. (2011), Williams, Colin P. (ed.), "Quantum Gates", Explorations in Quantum Computing, Texts in Computer Science, London: Springer, pp. 51–122, doi:10.1007/978-1-84628-887-6_2, ISBN 978-1-84628-887-6, retrieved 2021-05-14
  13. ^ Aharonov, Dorit (2003-01-09). "A Simple Proof that Toffoli and Hadamard are Quantum Universal". arXiv:quant-ph/0301040.
  14. ^ "Monroe Conference" (PDF). online.kitp.ucsb.edu.
  15. ^ "Demonstration of a small programmable quantum computer with atomic qubits" (PDF). Retrieved 2019-02-10.
  16. ^ Jones, Jonathan A. (2003). "Robust Ising gates for practical quantum computation". Physical Review A. 67 (1): 012317. arXiv:quant-ph/0209049. Bibcode:2003PhRvA..67a2317J. doi:10.1103/PhysRevA.67.012317. S2CID 119421647.
  17. ^ Rasmussen, S. E.; Zinner, N. T. (2020-07-17). "Simple implementation of high fidelity controlled- i swap gates and quantum circuit exponentiation of non-Hermitian gates". Physical Review Research. 2 (3): 033097. arXiv:2002.11728. Bibcode:2020PhRvR...2c3097R. doi:10.1103/PhysRevResearch.2.033097. ISSN 2643-1564.
  18. ^ Schuch, Norbert; Siewert, Jens (2003-03-10). "Natural two-qubit gate for quantum computation using the XY interaction". Physical Review A. 67 (3): 032301. arXiv:quant-ph/0209035. Bibcode:2003PhRvA..67c2301S. doi:10.1103/PhysRevA.67.032301. ISSN 1050-2947.
  19. ^ Dallaire-Demers, Pierre-Luc; Wilhelm, Frank K. (2016-12-05). "Quantum gates and architecture for the quantum simulation of the Fermi-Hubbard model". Physical Review A. 94 (6): 062304. arXiv:1606.00208. Bibcode:2016PhRvA..94f2304D. doi:10.1103/PhysRevA.94.062304. ISSN 2469-9926.
  20. ^ Shi, Xiao-Feng (2018-05-22). "Deutsch, Toffoli, and cnot Gates via Rydberg Blockade of Neutral Atoms". Physical Review Applied. 9 (5): 051001. arXiv:1710.01859. Bibcode:2018PhRvP...9e1001S. doi:10.1103/PhysRevApplied.9.051001. ISSN 2331-7019.
  21. ^ Aharonov, Dorit (2003-01-09). "A Simple Proof that Toffoli and Hadamard are Quantum Universal". arXiv:quant-ph/0301040.
  22. ^ Deutsch, David (September 8, 1989), "Quantum computational networks", Proc. R. Soc. Lond. A, 425 (1989): 73–90, Bibcode:1989RSPSA.425...73D, doi:10.1098/rspa.1989.0099, S2CID 123073680
  23. ^ Bausch, Johannes; Piddock, Stephen (2017), "The Complexity of Translationally-Invariant Low-Dimensional Spin Lattices in 3D", Journal of Mathematical Physics, 58 (11): 111901, arXiv:1702.08830, Bibcode:2017JMP....58k1901B, doi:10.1063/1.5011338, S2CID 8502985
  24. ^ Raz, Ran (2002). "On the complexity of matrix product". Proceedings of the Thirty-fourth Annual ACM Symposium on Theory of Computing: 144. doi:10.1145/509907.509932. ISBN 1581134959. S2CID 9582328.
  25. ^ Defining adjoined operators in Microsoft Q#
  26. ^ a b c Ömer, Bernhard (2000-01-20). Quantum Programming in QCL (PDF) (Thesis). Retrieved 2021-05-24.
  27. ^ Griffiths, D.J. (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. pp. 115–121, 126. ISBN 978-3-527-40601-2.
  28. ^ David Albert (1994). Quantum mechanics and experience. Harvard University Press. p. 35. ISBN 0-674-74113-7.
  29. ^ Sean M. Carroll (2019). Spacetime and geometry: An introduction to general relativity. Cambridge University Press. pp. 376–394. ISBN 978-1-108-48839-6.
  30. ^ David Wallace (2012). The emergent multiverse: Quantum theory according to the Everett Interpretation. Oxford University Press. ISBN 9780199546961.
  31. ^ Sean M. Carroll (2019). Something deeply hidden: Quantum worlds and the emergence of spacetime. Penguin Random House. ISBN 9781524743017.
  32. ^ Q# Online manual: Measurement
  33. ^ Juan Yin, Yuan Cao, Yu-Huai Li, et.c. (2017). "Satellite-based entanglement distribution over 1200 kilometers". Quantum Optics. 356: 1140–1144. arXiv:1707.01339.CS1 maint: uses authors parameter (link)
  34. ^ Billings, Lee. "China Shatters "Spooky Action at a Distance" Record, Preps for Quantum Internet". Scientific American.
  35. ^ Popkin, Gabriel (15 June 2017). "China's quantum satellite achieves 'spooky action' at record distance". Science - AAAS.
  36. ^ Dawson, Christopher M.; Nielsen, Michael (2006-01-01). "The Solovay-Kitaev algorithm". Quantum Information & Computation. 6 (1). Section 5.1, equation 23. arXiv:quant-ph/0505030. doi:10.26421/QIC6.1-6.
  37. ^ Matteo, Olivia Di (2016). "Parallelizing quantum circuit synthesis". Quantum Science and Technology. 1 (1): 015003. arXiv:1606.07413. Bibcode:2016QS&T....1a5003D. doi:10.1088/2058-9565/1/1/015003. S2CID 62819073.
  38. ^ Aaronson, Scott (2002). "Quantum Lower Bound for Recursive Fourier Sampling". Quantum Information and Computation. 3 (2): 165–174. arXiv:quant-ph/0209060. Bibcode:2002quant.ph..9060A. doi:10.26421/QIC3.2-7.
  39. ^ Q# online manual: Quantum Memory Management
  40. ^ Ryo, Asaka; Kazumitsu, Sakai; Ryoko, Yahagi (2020). "Quantum circuit for the fast Fourier transform". Quantum Information Processing. 19 (277): 277. arXiv:1911.03055. Bibcode:2020QuIP...19..277A. doi:10.1007/s11128-020-02776-5. S2CID 207847474.
  41. ^ Montaser, Rasha (2019). "New Design of Reversible Full Adder/Subtractor using R gate". International Journal of Theoretical Physics. 58 (1): 167–183. arXiv:1708.00306. Bibcode:2019IJTP...58..167M. doi:10.1007/s10773-018-3921-1. S2CID 24590164.
  42. ^ QCL 0.6.4 source code, the file "lib/examples.qcl"
  43. ^ S. J. Pauka, K. Das, R. Kalra, A. Moini, Y. Yang, M. Trainer, A. Bousquet, C. Cantaloube, N. Dick, G. C. Gardner, M. J. (2021). "A cryogenic CMOS chip for generating control signals for multiple qubits". Nature Electronics. 4 (4): 64–70. arXiv:1912.01299. doi:10.1038/s41928-020-00528-y.CS1 maint: uses authors parameter (link)

SourcesEdit