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In quantum information theory, a **quantum circuit** is a model for quantum computation in which a computation is a sequence of quantum gates, which are reversible transformations on a quantum mechanical analog of an *n*-bit register. This analogous structure is referred to as an *n*-qubit register.

The elementary logic gates of a classical computer, other than the NOT gate, are not reversible. Thus, for instance, for an AND gate one cannot always recover the two input bits from the output bit; for example, if the output bit is 0, we cannot tell from this whether the input bits are 0,1 or 1,0 or 0,0.

However, reversible gates in classical computers are easily constructed for bit strings of any length; moreover, these are actually of practical interest, since irreversible gates must always increase physical entropy. A reversible gate is a reversible function on *n*-bit data that returns *n*-bit data, where an *n*-bit data is a string of bits *x*_{1},*x*_{2}, ...,*x*_{n} of length *n*. The set of *n*-bit data is the space {0,1}^{n}, which consists of 2^{n} strings of 0's and 1's.

More precisely: an *n*-bit reversible gate is a bijective mapping *f* from the set {0,1}^{n} of *n*-bit data onto itself. An example of such a reversible gate *f* is a mapping that applies a fixed permutation to its inputs. For reasons of practical engineering, one typically studies gates only for small values of *n*, e.g. *n*=1, *n*=2 or *n*=3. These gates can be easily described by tables.

To define quantum gates, we first need to specify the quantum replacement of an *n*-bit datum. The *quantized version* of classical *n*-bit space {0,1}^{n} is the Hilbert space

This is by definition the space of complex-valued functions on {0,1}^{n} and is naturally an inner product space. This space can also be regarded as consisting of linear superpositions of classical bit strings. Note that *H*_{QB(n)} is a vector space over the complex numbers of dimension 2^{n}. The elements of this space are called *n*-qubits.

Using Dirac ket notation, if *x*_{1},*x*_{2}, ...,*x*_{n} is a classical bit string, then

is a special *n*-qubit corresponding to the function which maps this classical bit string to 1 and maps all other bit strings to 0; these 2^{n} special *n*-qubits are called *computational basis states*. All *n*-qubits are complex linear combinations of these computational basis states.

Quantum logic gates, in contrast to classical logic gates, are always reversible. One requires a special kind of reversible function, namely a unitary mapping, that is, a linear transformation of a complex inner product space that preserves the Hermitian inner product. An *n*-qubit (reversible) quantum gate is a unitary mapping *U* from the space *H*_{QB(n)} of *n*-qubits onto itself.

Typically, we are only interested in gates for small values of *n*.

A reversible *n*-bit classical logic gate gives rise to a reversible *n*-bit quantum gate as follows: to each reversible *n*-bit logic gate *f* corresponds a quantum gate *W*_{f} defined as follows:

Note that *W*_{f} permutes the computational basis states.

Of particular importance is the controlled NOT gate (also called CNOT gate) *W*_{CNOT} defined on a quantized 2 qubit. Other examples of quantum logic gates derived from classical ones are the Toffoli gate and the Fredkin gate.

However, the Hilbert-space structure of the qubits permits many quantum gates that are not induced by classical ones. For example, a relative phase shift is a 1 qubit gate given by multiplication by the unitary matrix:

so

Again, we consider first *reversible* classical computation. Conceptually, there is no difference between a reversible *n*-bit circuit and a reversible *n*-bit logic gate: either one is just an invertible function on the space of *n* bit data. However, as mentioned in the previous section, for engineering reasons we would like to have a small number of simple reversible gates, that can be put together to assemble any reversible circuit.

To explain this assembly process, suppose we have a reversible *n*-bit gate *f* and a reversible *m*-bit gate *g*. Putting them together means producing a new circuit by connecting some set of *k* outputs of *f* to some set of *k* inputs of *g* as in the figure below. In that figure, *n*=5, *k*=3 and *m*=7. The resulting circuit is also reversible and operates on *n*+*m*−*k* bits.

We will refer to this scheme as a *classical assemblage* (This concept corresponds to a technical definition in Kitaev's pioneering paper cited below). In composing these reversible machines, it is important to ensure that the intermediate machines are also reversible. This condition assures that *intermediate* "garbage" is not created (the net physical effect would be to increase entropy, which is one of the motivations for going through this exercise).

Now it is possible to show that the Toffoli gate is a universal gate. This means that given any reversible classical *n*-bit circuit *h*, we can construct a classical assemblage of Toffoli gates in the above manner to produce an (*n*+*m*)-bit circuit *f* such that

where there are *m* underbraced zeroed inputs and

- .

Notice that the end result always has a string of *m* zeros as the ancilla bits. No "rubbish" is ever produced, and so this computation is indeed one that, in a physical sense, generates no entropy. This issue is carefully discussed in Kitaev's article.

More generally, any function *f* (bijective or not) can be simulated by a circuit of Toffoli gates. Obviously, if the mapping fails to be injective, at some point in the simulation (for example as the last step) some "garbage" has to be produced.

For quantum circuits a similar composition of qubit gates can be defined. That is, associated to any *classical assemblage* as above, we can produce a reversible quantum circuit when in place of *f* we have an *n*-qubit gate *U* and in place of *g* we have an *m*-qubit gate *W*. See illustration below:

The fact that connecting gates this way gives rise to a unitary mapping on *n*+*m*−*k* qubit space is easy to check. In a real quantum computer the physical connection between the gates is a major engineering challenge, since it is one of the places where decoherence may occur.

There are also universality theorems for certain sets of well-known gates; such a universality theorem exists, for instance, for the pair consisting of the single qubit phase gate *U*_{θ} mentioned above (for a suitable value of θ), together with the 2-qubit CNOT gate *W*_{CNOT}. However, the universality theorem for the quantum case is somewhat weaker than the one for the classical case; it asserts only that any reversible *n*-qubit circuit can be *approximated* arbitrarily well by circuits assembled from these two elementary gates. Note that there are uncountably many possible single qubit phase gates, one for every possible angle θ, so they cannot all be represented by a finite circuit constructed from {*U*_{θ}, *W*_{CNOT})}.

So far we have not shown how quantum circuits are used to perform computations. Since many important numerical problems reduce to computing a unitary transformation *U* on a finite-dimensional space (the celebrated discrete Fourier transform being a prime example), one might expect that some quantum circuit could be designed to carry out the transformation *U*. In principle, one needs only to prepare an *n* qubit state ψ as an appropriate superposition of computational basis states for the input and measure the output *U*ψ. Unfortunately, there are two problems with this:

- One cannot measure the phase of ψ at any computational basis state so there is no way of reading out the complete answer. This is in the nature of measurement in quantum mechanics.
- There is no way to efficiently prepare the input state ψ.

This does not prevent quantum circuits for the discrete Fourier transform from being used as intermediate steps in other quantum circuits, but the use is more subtle. In fact quantum computations are *probabilistic*.

We now provide a mathematical model for how quantum circuits can simulate *probabilistic* but classical computations. Consider an *r*-qubit circuit *U* with register space *H*_{QB(r)}. *U* is thus a unitary map

In order to associate this circuit to a classical mapping on bitstrings, we specify

- An
*input register**X*= {0,1}^{m}of*m*(classical) bits. - An
*output register**Y*= {0,1}^{n}of*n*(classical) bits.

The contents *x* = *x*_{1}, ..., *x*_{m} of the classical input register are used to initialize the qubit register in some way. Ideally, this would be done with the computational basis state

where there are *r*-*m* underbraced zeroed inputs. Nevertheless, this perfect initialization is completely unrealistic. Let us assume therefore that the initialization is a mixed state given by some density operator *S* which is near the idealized input in some appropriate metric, e.g.

Similarly, the output register space is related to the qubit register, by a *Y* valued observable *A*. Note that observables in quantum mechanics are usually defined in terms of *projection valued measures* on **R**; if the variable happens to be discrete, the projection valued measure reduces to a family {E_{λ}} indexed on some parameter λ ranging over a countable set. Similarly, a *Y* valued observable, can be associated with a family of pairwise orthogonal projections {E_{y}} indexed by elements of *Y*. such that

Given a mixed state *S*, there corresponds a probability measure on *Y* given by

The function *F*:*X* → *Y* is computed by a circuit *U*:*H*_{QB(r)} → *H*_{QB(r)} to within ε if and only if for all bitstrings *x* of length *m*

Now

so that

**Theorem**. If ε + δ < 1/2, then the probability distribution

on *Y* can be used to determine *F*(*x*) with an arbitrarily small probability of error by majority sampling, for a sufficiently large sample size. Specifically, take *k* independent samples from the probability distribution Pr on *Y* and choose a value on which more than half of the samples agree. The probability that the value *F*(*x*) is sampled more than *k*/2 times is at least

where γ = 1/2 - ε - δ.

This follows by applying the Chernoff bound.

In mathematical physics and mathematics, the **Pauli matrices** are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma, they are occasionally denoted by tau when used in connection with isospin symmetries. They are

In quantum computing, a **qubit** or **quantum bit** is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include: the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property which is fundamental to quantum mechanics and quantum computing.

**Shor's algorithm** is a polynomial-time quantum computer algorithm for integer factorization. Informally, it solves the following problem: Given an integer , find its prime factors. It was invented in 1994 by the American mathematician Peter Shor.

**Grover's algorithm** is a quantum algorithm that finds with high probability the unique input to a black box function that produces a particular output value, using just evaluations of the function, where is the size of the function's domain. It was devised by Lov Grover in 1996.

In logic circuits, the **Toffoli gate**, invented by Tommaso Toffoli, is a universal reversible logic gate, which means that any reversible circuit can be constructed from Toffoli gates. It is also known as the "controlled-controlled-not" gate, which describes its action. It has 3-bit inputs and outputs; if the first two bits are both set to 1, it inverts the third bit, otherwise all bits stay the same.

The **Deutsch–Jozsa algorithm** is a quantum algorithm, proposed by David Deutsch and Richard Jozsa in 1992 with improvements by Richard Cleve, Artur Ekert, Chiara Macchiavello, and Michele Mosca in 1998. Although of little practical use, it is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm and is the inspiration for Simon's Algorithm which is, in turn, the inspiration for Shor's Algorithm. It is also a deterministic algorithm, meaning that it always produces an answer, and that answer is always correct.

In quantum mechanics, the **Bloch sphere** is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.

In quantum computing and specifically the quantum circuit model of computation, a **quantum logic 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.

**Quantum error correction** (**QEC**) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.

The **Bell states**, a concept in quantum information science, are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. The Bell states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1: . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will collapse it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will assign one of two possible values to the other qubit instantly, where the value assigned depends on which Bell state the two qubits are in. Bell states can be generalized to represent specific quantum states of multi-qubit systems, such as the GHZ state for 3 subsystems.

In quantum information theory, **superdense coding** is a quantum communication protocol to transmit two classical bits of information from a sender to a receiver, by sending only one qubit from Alice to Bob, under the assumption of Alice and Bob pre-sharing an entangled state. This protocol was first proposed by Bennett and Wiesner in 1992 and experimentally actualized in 1996 by Mattle, Weinfurter, Kwiat and Zeilinger using entangled photon pairs. By performing one of four quantum gate operations on the (entangled) qubit she possesses, Alice can prearrange the measurement Bob makes. After receiving Alice's qubit, operating on the pair and measuring both, Bob has two classical bits of information. If Alice and Bob do not already share entanglement before the protocol begins, then it is impossible to send two classical bits using 1 qubit, as this would violate Holevo's theorem.

In computing science, the **controlled NOT gate** is a quantum logic gate that is an essential component in the construction of a gate-based quantum computer. It can be used to entangle and disentangle EPR states. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations.

In quantum mechanics, notably in quantum information theory, **fidelity** is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.

**Entanglement distillation** is the transformation of *N* copies of an arbitrary entangled state into some number of approximately pure Bell pairs, using only *local operations and classical communication* (LOCC).

The **Quantum phase estimation algorithm**, is a quantum algorithm to estimate the phase of an eigenvector of a unitary operator. More precisely, given a unitary matrix and a quantum state such that , the algorithm estimates the value of with high probability within additive error , using controlled-U operations.

In quantum computing, the **quantum Fourier transform** is a linear transformation on quantum bits, and is the quantum analogue of the inverse discrete Fourier transform. The quantum Fourier transform is a part of many quantum algorithms, notably Shor's algorithm for factoring and computing the discrete logarithm, the quantum phase estimation algorithm for estimating the eigenvalues of a unitary operator, and algorithms for the hidden subgroup problem. The quantum Fourier transform was invented by Don Coppersmith.

**Linear Optical Quantum Computing** or **Linear Optics Quantum Computation** (**LOQC**) is a paradigm of quantum computation, allowing universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements, or optical instruments to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.

The **KLM scheme** or **KLM protocol** is an implementation of linear optical quantum computing (LOQC), developed in 2000 by Knill, Laflamme and Milburn. This protocol makes it possible to create universal quantum computers solely with linear optical tools. The KLM protocol uses linear optical elements, single photon sources and photon detectors as resources to construct a quantum computation scheme involving only ancilla resources, quantum teleportations and error corrections.

**Quantum counting algorithm** is a quantum algorithm for efficiently counting the number of solutions for a given search problem. The algorithm is based on the quantum phase estimation algorithm and on Grover's search algorithm.

The **Bernstein–Vazirani algorithm**, which solves the **Bernstein–Vazirani problem** is a quantum algorithm invented by Ethan Bernstein and Umesh Vazirani in 1992. It's a restricted version of the Deutsch–Jozsa algorithm where instead of distinguishing between two different classes of functions, it tries to learn a string encoded in a function. The Bernstein–Vazirani algorithm was designed to prove an oracle separation between complexity classes BQP and BPP.

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- Q-circuit is a macro package for drawing quantum circuit diagrams in LaTeX.
- Quantum Circuit Simulator (Davy Wybiral) a browser-based quantum circuit diagram editor and simulator.
- Quantum Computing Playground a browser-based quantum scripting environment.
- Quirk - Quantum Circuit Toy a browser-based quantum circuit diagram editor and simulator.

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