# Quantum circuit

Last updated

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 graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation.

## Reversible classical logic gates

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 01 or 10 or 00.

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 x1,x2, ...,xn of length n. The set of n-bit data is the space {0,1}n, which consists of 2n 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.

## Quantum logic gates

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

$H_{\operatorname {QB} (n)}=\ell ^{2}(\{0,1\}^{n}).$ 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 HQB(n) is a vector space over the complex numbers of dimension 2n. The elements of this space are called n-qubits.

Using Dirac ket notation, if x1,x2, ...,xn is a classical bit string, then

$|x_{1},x_{2},\cdots ,x_{n}\rangle \quad$ 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 2n 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 HQB(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 Wf defined as follows:

$W_{f}(|x_{1},x_{2},\cdots ,x_{n}\rangle )=|f(x_{1},x_{2},\cdots ,x_{n})\rangle .$ Note that Wf permutes the computational basis states.

Of particular importance is the controlled NOT gate (also called CNOT gate) WCNOT 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 phase shift operation:

$P(\varphi )={\begin{bmatrix}1&0\\0&e^{i\varphi }\end{bmatrix}},$ so

$P(\varphi )|0\rangle =|0\rangle \quad P(\varphi )|1\rangle =e^{i\varphi }|1\rangle .$ ## Reversible logic circuits

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+mk 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).

Note that each horizontal line on the above picture represents either 0 or 1, not these probabilities. Since quantum computations are reversible, at each 'step' the number of lines must be the same number of input lines. Also, each input combination must be mapped to a single combination at each 'step'. This means that each intermediate combination in a quantum circuit is a bijective function of the input. 

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

$f(x_{1},\ldots ,x_{n},\underbrace {0,\dots ,0} )=(y_{1},\ldots ,y_{n},\underbrace {0,\ldots ,0} )$ where there are m underbraced zeroed inputs and

$(y_{1},\ldots ,y_{n})=h(x_{1},\ldots ,x_{n})$ .

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+mk 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 Uθ mentioned above (for a suitable value of θ), together with the 2-qubit CNOT gate WCNOT. 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 {Uθ, WCNOT}.

## Quantum computations

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 Uψ. 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 HQB(r). U is thus a unitary map

$H_{\operatorname {QB} (r)}\rightarrow H_{\operatorname {QB} (r)}.$ In order to associate this circuit to a classical mapping on bitstrings, we specify

• An input registerX = {0,1}m of m (classical) bits.
• An output registerY = {0,1}n of n (classical) bits.

The contents x = x1, ..., xm 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

$|{\vec {x}},0\rangle =|x_{1},x_{2},\cdots ,x_{m},\underbrace {0,\dots ,0} \rangle ,$ 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.

$\operatorname {Tr} \left({\big |}|{\vec {x}},0\rangle \langle {\vec {x}},0|-S{\big |}\right)\leq \delta .$ 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 {Eλ} 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 {Ey} indexed by elements of Y. such that

$I=\sum _{y\in Y}\operatorname {E} _{y}.$ Given a mixed state S, there corresponds a probability measure on Y given by

$\operatorname {Pr} \{y\}=\operatorname {Tr} (S\operatorname {E} _{y}).$ The function F:XY is computed by a circuit U:HQB(r)HQB(r) to within ε if and only if for all bitstrings x of length m

$\left\langle {\vec {x}},0{\big |}U^{*}\operatorname {E} _{F(x)}U{\big |}{\vec {x}},0\right\rangle =\left\langle \operatorname {E} _{F(x)}U(|{\vec {x}},0\rangle ){\big |}U(|{\vec {x}},0\rangle )\right\rangle \geq 1-\epsilon .$ Now

$\left|\operatorname {Tr} (SU^{*}\operatorname {E} _{F(x)}U)-\left\langle {\vec {x}},0{\big |}U^{*}\operatorname {E} _{F(x)}U{\big |}{\vec {x}},0\right\rangle \right|\leq \operatorname {Tr} ({\big |}|{\vec {x}},0\rangle \langle {\vec {x}},0|-S{\big |})\|U^{*}\operatorname {E} _{F(x)}U\|\leq \delta$ so that

$\operatorname {Tr} (SU^{*}\operatorname {E} _{F(x)}U)\geq 1-\epsilon -\delta .$ Theorem. If ε + δ < 1/2, then the probability distribution

$\operatorname {Pr} \{y\}=\operatorname {Tr} (SU^{*}\operatorname {E} _{y}U)$ 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

$1-e^{-2\gamma ^{2}k},$ where γ = 1/2 - ε - δ.

This follows by applying the Chernoff bound.

## Related Research Articles

In computational complexity theory, bounded-error quantum polynomial time (BQP) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. It is the quantum analogue to the complexity class BPP. Quantum computing is the exploitation of collective properties of quantum states, such as superposition and entanglement, to perform computation. The devices that perform quantum computations are known as quantum computers. They are believed to be able to solve certain computational problems, such as integer factorization, substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. Expansion is expected in the next few years as the field shifts toward real-world use in pharmaceutical, data security and other applications.

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Moreover, the sender may not know the location of the recipient, and does not know which particular quantum state will be transferred.

In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classic 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 that 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 discovered in 1994 by the American mathematician Peter Shor.

The Deutsch–Jozsa algorithm is a deterministic 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 current practical use, it is one of the first examples of a quantum algorithm that is exponentially faster than any possible deterministic classical algorithm.

In quantum physics, a measurement is the testing or manipulation of a physical system to yield a numerical result. The predictions that quantum physics makes are in general probabilistic. The mathematical tools for making predictions about what measurement outcomes may occur were developed during the 20th century and make use of linear algebra and functional analysis. In quantum mechanics and computing, 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 or EPR pairs are specific quantum states of two qubits that represent the simplest examples of quantum entanglement; conceptually, they fall under the study of quantum information science. 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 or more subsystems. In quantum information theory, superdense coding is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assumption of sender and received pre-sharing an entangled resource. In its simplest form, the protocol involves two parties, often referred to as Alice and Bob in this context, which share a pair of maximally entangled qubits, and allows Alice to transmit two bits to Bob by sending only one qubit. This protocol was first proposed by Bennett and Wiesner in 1970 and experimentally actualized in 1996 by Mattle, Weinfurter, Kwiat and Zeilinger using entangled photon pairs. Superdense coding can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair. In computer 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 Bell states. Any quantum circuit can be simulated to an arbitrary degree of accuracy using a combination of CNOT gates and single qubit rotations.

In computational complexity theory, PostBQP is a complexity class consisting of all of the computational problems solvable in polynomial time on a quantum Turing machine with postselection and bounded error.

Holevo's theorem is an important limitative theorem in quantum computing, an interdisciplinary field of physics and computer science. It is sometimes called Holevo's bound, since it establishes an upper bound to the amount of information that can be known about a quantum state. It was published by Alexander Holevo in 1973.

Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems in relation to these complexity classes, as well as the relationship between quantum complexity classes and classical complexity classes.

Quantum refereed game in quantum information processing is a class of games in the general theory of quantum games. It is played between two players, Alice and Bob, and arbitrated by a referee. The referee outputs the pay-off for the players after interacting with them for a fixed number of rounds, while exchanging quantum information.

Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best solution to a problem from a set of possible solutions. Mostly, the optimization problem is formulated as a minimization problem, where one tries to minimize an error which depends on the solution: the optimal solution has the minimal error. Different optimization techniques are applied in various fields such as mechanics, economics and engineering, and as the complexity and amount of data involved rise, more efficient ways of solving optimization problems are needed. The power of quantum computing may allow problems which are not practically feasible on classical computers to be solved, or suggest a considerable speed up with respect to the best known classical algorithm.

In quantum information and computation, the Solovay–Kitaev theorem says, roughly, that if a set of single-qubit quantum gates generates a dense subset of SU(2) then that set is guaranteed to fill SU(2) quickly, which means any desired gate can be approximated by a fairly short sequence of gates from the generating set. Robert M. Solovay initially announced the result on an email list in 1995, and Alexei Kitaev independently gave an outline of its proof in 1997. Christopher M. Dawson and Michael Nielsen call the theorem one of the most important fundamental results in the field of quantum computation. 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.

1. "Introduction to the Quantum Circuit Model" (PDF).