quantum circuit

Most elementary logic gates of a classical computer 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, ...,x_n 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.

The quantum logic gates are reversible unitary transformations on at least one qubit. Multiple qubits taken together are referred to as quantum registers. 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

:$H\_\{\backslash operatorname\{QB\}(n)\}=\; \backslash ell^2(\backslash \{0,1\backslash \}^n).$

This is by definition the space of complex-valued functions on {0,1}ⁿ and is naturally an inner product space. $\backslash ell^2$ means the function is a square-integrable function. This space can also be regarded as consisting of linear combinations, or superpositions, of classical bit strings. Note that H_QB(n) is a vector space over the complex numbers of dimension 2ⁿ. The elements of this vector space are the possible state-vectors of n-qubit quantum registers.

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

:$|\; x\_1,\; x\_2,\; \backslash cdots,x\_n\; \backslash rangle\; \backslash quad$

is a special n-qubit register corresponding to the function which maps this classical bit string to 1 and maps all other bit strings to 0; these 2ⁿ special n-qubit registers are called computational basis states. All n-qubit registers 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-qubit registers 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:

:$W\_f(\; |\; x\_1,\; x\_2,\; \backslash cdots,x\_n\; \backslash rangle)\; =\; |f(x\_1,\; x\_2,\; \backslash cdots,\; x\_n)\; \backslash rangle.$

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 phase shift operator:

:

so

:$P(\backslash varphi)|\; 0\; \backslash rangle\; =\; |\; 0\; \backslash rangle\; \backslash quad\; P(\backslash varphi)|\; 1\; \backslash rangle\; =\; e^\{i\backslash varphi\}|\; 1\; \backslash rangle.$

{{main|reversible computing}}

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.

Image:Reversible circuit composition.svg

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.{{cite web|url=https://www.cs.cmu.edu/~odonnell/quantum15/QuantumComputationScribeNotesByRyanODonnellAndJohnWright.pdf|title=Introduction to the Quantum Circuit Model}}

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

:$f(x\_1,\; \backslash ldots,\; x\_n,\; \backslash underbrace\{0,\; \backslash dots,\; 0\})\; =\; (y\_1,\; \backslash ldots,\; y\_n,\; \backslash underbrace\{0,\; \backslash ldots\; ,\; 0\})$

where there are m underbraced zeroed inputs and

:$(y\_1,\; \backslash ldots,\; y\_n)\; =\; h(x\_1,\; \backslash ldots,\; x\_n)$.

Notice that the 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:

Image:Quantum circuit composition.svg

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 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 H_QB(r). U is thus a unitary map

:$H\_\{\backslash operatorname\{QB\}(r)\}\; \backslash rightarrow$

H_{\operatorname{QB}(r)}.

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 = x₁, ..., 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

:$|\backslash vec\{x\},0\backslash rangle=\; |\; x\_1,\; x\_2,\; \backslash cdots,\; x\_\{m\},\; \backslash underbrace\{0,\; \backslash dots,\; 0\}\; \backslash rangle,$

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.

:$\backslash operatorname\{Tr\}\backslash left(\backslash big||\backslash vec\{x\},0\backslash rangle\; \backslash langle\; \backslash vec\{x\},0\; |\; -\; S\backslash big|\backslash right)\; \backslash leq\; \backslash delta.$

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 {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

:$I\; =\; \backslash sum\_\{y\; \backslash in\; Y\}\; \backslash operatorname\{E\}\_y.$

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

given by

:$\backslash operatorname\{Pr\}\backslash \{y\backslash \}\; =\; \backslash operatorname\{Tr\}(S\; \backslash operatorname\{E\}\_y\; ).$

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

:$\backslash left\backslash langle\; \backslash vec\{x\},0\; \backslash big|\; U^*\; \backslash operatorname\{E\}\_\{F(x)\}\; U$

\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

:$\backslash left|\; \backslash operatorname\{Tr\}\; (S\; U^*\; \backslash operatorname\{E\}\_\{F(x)\}\; U)\; -\; \backslash left\backslash langle\; \backslash vec\{x\},0\; \backslash big|\; U^*\; \backslash operatorname\{E\}\_\{F(x)\}\; U$

\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

:$\backslash operatorname\{Tr\}\; (S\; U^*\; \backslash operatorname\{E\}\_\{F(x)\}\; U)\; \backslash geq\; 1\; -\; \backslash epsilon\; -\; \backslash delta.$

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

:$\backslash operatorname\{Pr\}\backslash \{y\backslash \}\; =\; \backslash operatorname\{Tr\}\; (S\; U^*\; \backslash operatorname\{E\}\_\{y\}\; U)$

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

:$1\; -\; e^\{-\; 2\; \backslash gamma^2\; k\},$

where γ = 1/2 - ε - δ.

This follows by applying the Chernoff bound.

With the advent of quantum computing, there has been a significant surge in both the number of developers and available tools.{{cite web | url=https://medium.com/@josephmeng96/accelerating-quantum-simulations-w-fpgas-84b09569ad0f | title=Accelerating Quantum Simulations w/ FPGAs | date=19 August 2020 }} However, the slow pace of technological advancement and the high maintenance costs associated with quantum computers have limited broader participation in this field. In response, developers have turned to simulators, such as IBM's Qiskit, to model quantum behavior without relying solely on real quantum hardware. Nevertheless, simulators, being classical computers, are constrained by computation speed. The fundamental advantage of quantum computers lies in their ability to process qubits, leveraging properties like entanglement and superposition simultaneously. By running quantum simulations on classical computers, the inherent parallelism of quantum computing is taken away. Moreover, as the number of simulated qubits increases, the simulation's speed decreases proportionally.

In a quantum circuit, the vectors are used to represent the state of the qubits and different matrices are used to represent the gate that is applied on the qubits. Since linear algebra is a major component of the quantum simulation, Field Programmable Gate Arrays (FPGAs) could be used to accelerate the simulation of quantum computing. FPGA is a kind of hardware that excels at executing operations in parallel, supports pipelining, has on-chip memory resources with low access latency, and offers the flexibility to reconfigure the hardware architecture on-the-fly which make it a well suited tool to handle matrix multiplication.

The main idea of accelerating quantum computing simulations is to offload some of the heavy computation to special hardware like FPGA in order to speed up the whole simulation process. And the bigger quantum circuits (more qubits and more gates) we simulate, the more speedup we gain from offloading to FPGA compared with software simulations on CPU. The data flow of the simulation is explained below. First, the user inputs all the information of the quantum circuit including initial state and various gates through the user interface. Then, all this information is compressed and sent to the FPGA through some hardware communication protocols like AXI. Then, all the information is stored in the on-chip memory in the FPGA. And the simulation starts when the data is read from the memory and sent to the Matrix multiplication module. After all the calculation is done, the result will be sent back to the memory and to the CPU.

Suppose we are simulating 5-qubit circuits, then we need to store the vector that holds 32 (2⁵) 16-bit values, each of which represents the square-root probability of a possible existing state. We also need to store the 32x32 matrix that represents the gate. In order to parallel this computation, we can store the 32 rows of the matrix separately and replicate 32 row_vec_mult hardware such that each row can calculate the multiplication in parallel. This will dramatically speed up the simulation with a price of more hardware and memory usage in FPGA. {{cite web | url=https://medium.com/@josephmeng96/accelerating-quantum-simulations-w-fpgas-84b09569ad0f | title=Accelerating Quantum Simulations w/ FPGAs | date=19 August 2020 }}

It has been discovered that with careful hardware design, it's possible to achieve a hardware architecture with O(n) time complexity, where 'n' denotes the number of qubits. In contrast, the runtime of Numpy approaches O(2^2^n). This finding underscores the feasibility of leveraging FPGAs to accelerate quantum computing simulations. {{cite web | url=https://medium.com/@josephmeng96/accelerating-quantum-simulations-w-fpgas-84b09569ad0f | title=Accelerating Quantum Simulations w/ FPGAs | date=19 August 2020 }}

{{Commons category |Quantum circuit}}

• Abstract index notation
• Angular momentum diagrams (quantum mechanics)
• Circuit complexity and BQP
• Matrix product state uses Penrose graphical notation
• Quantum register
• Spin networks
• Trace diagram

{{Reflist}}

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• [https://cquic.unm.edu/resources/ Q-circuit] {{Webarchive|url=https://web.archive.org/web/20190323101451/https://cquic.unm.edu/resources/ |date=2019-03-23 }} is a macro package for drawing quantum circuit diagrams in LaTeX.
• [https://qcsimulator.github.io/ Quantum Circuit Simulator (Davy Wybiral)] ({{GitHub|qcsimulator/qcsimulator.github.io}}) a browser-based quantum circuit diagram editor and simulator.
• [https://www.quantumplayground.net/ Quantum Computing Playground] ({{GitHub|gwroblew/Quantum-Computing-Playground}}) a browser-based quantum scripting environment.
• [https://algassert.com/quirk Quirk - Quantum Circuit Toy] ({{GitHub|Strilanc/Quirk}}) a browser-based quantum circuit diagram editor and simulator.

{{Electronic components}}

{{Quantum computing}}

{{Quantum mechanics topics}}

{{emerging technologies|quantum=yes|other=yes}}

Category:Quantum information science

Category:Models of computation