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Quantum Computer

Quantum computers require systems that are well isolated from the decohering effects of their environment, while at the same time allowing precise manipulation during computation.

From: Optical Fiber Telecommunications VII, 2020

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Quantum Computing and Communication

Paul E. Black, ... Carl J. Williams, in Advances in Computers, 2002

Abstract

A quantum computer, if built, will be to an ordinary computer as a hydrogen bomb is to gunpowder, at least for some types of computations. Today no quantum computer exists, beyond laboratory prototypes capable of solving only tiny problems, and many practical problems remain to be solved. Yet the theory of quantum computing has advanced significantly in the past decade, and is becoming a significant discipline in itself. This article explains the concepts and basic mathematics behind quantum computers and some of the promising approaches for building them. We also discuss quantum communication, an essential component of future quantum information processing, and quantum cryptography, widely expected to be the first practical application for quantum information technology.

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Superconducting Nanomaterials

Donglu Shi, ... Nicholas Bedford, in Nanomaterials and Devices, 2015

8.5.1 Quantum Computers

The quantum computer is a class of physical devices that can perform high-speed mathematical and logical operations and that can store and process quantum information following the laws of quantum mechanics. When a device's processing and calculation are based on quantum information and run with quantum algorithms, it is a quantum computer. The concept of the quantum computer stems from research on reversible computers. The purpose is to address the issues regarding energy consumption of computers.

In the 1960s and 1970s, researchers discovered that energy consumption would lead to overheating of computer chips, which greatly affected the integration scale, thus limiting the ultimate speed of computers. Studies found that energy consumption comes from the irreversible operations in calculations. So, must calculation involve irreversible operations to complete? The answer is, for all the classic computers there is the possibility of a corresponding reversible computer with unchanged computing power. Because every step of the operation can be converted into a reversible operation, in quantum mechanics it can be expressed as a unitary transformation. The early quantum computer was a classical computer described by quantum mechanics languages and did not involve the use of the essential features of quantum mechanics, such as quantum superposition and coherence. In a classical computer, the basic information unit is a bit; the computing object is various bit sequences. Similarly, in a quantum computer, the basic information unit is the quantum bit, with operation targeting qubit sequences. The difference is that the quantum bit sequences not only are located at all orthogonal superposition states but also can be at the entangled state. In addition to the possibility of quantum parallel computing, these particular quantum states also brought about many wonderful features. In contrast to classical computers, quantum computers can perform arbitrary unitary transformation. Obtaining the output state, measurement can lead to the computing results. Thus, quantum computing generated a tremendous expansion of classical calculations. In the form of mathematics, classical computing can be seen as a special class of quantum computation. Quantum computers may transform a stack of each component; all of these transformations are completed at the same time and would stack at a certain rate of probability to give the results. This calculation is called quantum parallel computation. In addition to parallel computing, quantum computers have another important use, that is, to simulate quantum systems. This is a task beyond the competency of classical computers (Figure 8.3).

Figure 8.3. Comparison of traditional bits and qubits.

Whether using quantum parallel computing or quantum simulation computing, both are required to make use of quantum coherence. Unfortunately, it is difficult to maintain quantum coherence in a practical system. In quantum computers, quantum bits are not an isolated system. Rather, they will interact with the external environment, leading to the attenuation of quantum coherence, namely decoherence. So, the core issue in designing a quantum computer is to overcome decoherence. The discovery of quantum coding is by far the most effective way to overcome decoherence. The major quantum coding schemes include quantum error-correcting code, quantum error-avoiding code, and quantum error-prevention code. Quantum error-correcting code is the classic analog error-correcting code, which is a class of codes that attracted the most attention from researchers. Its merit lies in its wide application, but its drawback is its lower efficiency.

So far, there are still no quantum computers in the real sense in the world. However, many laboratories around the world are pursuing this goal with great enthusiasm. Many methods have been proposed regarding how to realize quantum computing. The problem lies in the fact that achieving the experimental manipulation of the micro-quantum state is very difficult. The currently available programs are mainly based on the interaction of atoms and cavity, as well as cold trap bound ions, electron or nuclear spin resonance, quantum dot manipulation, and superconducting quantum interference. It is difficult to say which option is more promising. From the perspective of integration and miniaturization, the quantum point program and superconducting Josephson junction program combined may be most appropriate. In the future, perhaps a novel design will emerge and make all the existing proposals obsolete; the new method might be based on some kind of novel material. Quantum computers are not intended to replace existing classical computers. The concept of quantum computers provides a new perspective for computation and is different from other new concepts of computers, such as optical computers and biocomputers. The role of quantum computers is far more than to solve some intractable problems that are possibly found in classical computers.

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QUANTUM OPTICS | Entanglement and Quantum Information

P.G. Kwiat, D.F.V. James, in Encyclopedia of Modern Optics, 2005

Quantum Computing

The most challenging and powerful application of quantum information is large-scale quantum computing. Two features that make quantum information processing potentially powerful are the exponentially large Hilbert space, which gives quantum registers very large capacities, and quantum parallelism, which means that data processing tasks can be performed very efficiently. One fundamental drawback, which severely constrains useful applications of quantum computers, is that the final measurement can only produce a number of classical bits equal to the number of qubits. Thus, quantum computers are limited to performing tasks in which a small amount of information is meant to be gleaned from processing a large amount of data; examples include searching an unstructured database for a specific entry (Grover's algorithm) or finding the periodicity of a function (the quantum Fourier transform). This second task is central to Shor's factor-finding algorithm, the most famous quantum computing algorithm to date.

A practicable quantum computer technology must have at least the following features, first identified by DiVincenzo:

•

A set of well-characterized, distinguishable qubits to form a quantum data storage register.

•

The ability to initialize the qubits of the register in a simple fiducial state.

•

Decoherence times that are much longer than the time needed to perform logical operations.

•

The ability to perform any single qubit operation on any qubit in the register, and the ability to perform two-qubit conditional logic gates (such as the CNOT gate: α|00 〉 +β|01〉 +γ|10〉 +δ|11〉→α|00〉 +β|01〉 +γ|11〉 +δ|10〉). Together, these operations constitute a universal set of quantum gates, from which all other gates can be synthesized.

•

The ability to measure each qubit.

All-optical schemes have been used to implement small quantum algorithms. However, most of the approaches are not scalable, due to the limitations of non-linear optics to perform a CNOT gate at the single-photon level. Recent proposals have suggested that very high efficiency single-photon detectors, along with sources of single photons 'on-demand,' allow scalable quantum computing with only linear optics; preliminary two-qubit gates have been experimentally demonstrated. A number of other promising candidate technologies that meet DiVincenzo's five requirements are being pursued vigorously. For example, the ability to create multiqubit entanglement and perform reliable measurements on trapped ions cooled and manipulated by lasers has recently been demonstrated. Solid state systems offer the possibility of scalability, and a number of schemes, such as quantum dots, isolated impurities with nuclear spins, and superconducting quantum interference devices (SQUIDs), are being investigated. It is possible that the final quantum computing technology may take the form of a hybrid between various present approaches.

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Nanoelectronics

D.J. Paul, in Encyclopedia of Physical Science and Technology (Third Edition), 2003

IX.D Quantum Computing

Quantum computing is a radically different architecture that uses the interference properties of entangle quantum mechanical particles to allow each bit of quantum information (called a qubit) in the computer to be intimately linked to every other qubit in the system. Quantum computing is therefore a massively parallel architecture at a level that is impossible in any classical architecture.

The important pieces of quantum mechanics for quantum computing is the Einstein–Podolsky–Rosen (EPR) paradox and the Bell inequality. If two particles are entangled and taken to the opposite ends of the universe, then quantum theory determines that if one measures the properties of one, you automatically then know the properties of the other since they are intimately linked. An example is the spin of a photon where if one has a right-hand polarization then the other after entanglement must have a left-hand polarization through quantum theory. Moreover, the state of one after a measurement automatically defines the state of the other. Before the measurement it is impossible to predict exactly which state each particle may be in. This is as if information has traveled instantaneously across the enormous distance of the universe, which is impossible from Einstein's special relatively theory.

A bit is a two-state system from a physical point of view. In quantum mechanics, if a bit has two distinguishable states, then it may also exist in a superposition of these two states called a qubit. If each of these superpositions are entangled so that there is a further superposition of each qubit, then the state of each qubit depends on the states of every other qubit. Gate operations on these qubits provide a time evolution of the system where every entangle particle is affected and so a massive parallel computation results. This enormous parallelism allows classically numerically intensive problems such as the traveling salesman problem (known as NP problems in mathematics) to be solved in real time that cannot be solved with 100 years of present supercomputing time. Potential applications are factorization for cryptography, fast database searching, and modeling of quantum systems.

To produce a quantum computer, there are five criteria called the DiVincenzo checklist that must be adhered to:

A scalable physical system with well characterized qubits.

Initialization of simple fiducial states. This basically is setting the qubits into similar quantum states before any calculation can proceed. An example is setting all the spins in one system to be identical.

Long relevant decoherence times, much longer than the gate operation. As the decoherence time dictates the length of time the qubits can be entangled without loss of any information, any computation must be finished before the qubits loose information.

A universal set of quantum gates. For a quantum computer, two types of gate operation are required to produce a universal computer that may be designed and programmed to complete any computation task. These are the single- and two-qubit operations, and from these a computer for factorization, database searching or quantum system simulation may be built and operated. These two-gate operations can be achieved by different techniques that depend on the specific two-state system used for each qubit.

A qubit-specific measurement capability. Once a calculation is complete, the information must be outputed from the qubits.

At present only a few systems with up to 5 or so qubits have been demonstrated. For most applications such as factorization of large numbers for cryptography, 30,000 qubits or so are required, although even 50 qubits will solve problems quickly that are impossibly slow on classical computers. Ion traps and nuclear magnetic resonance systems have so far demonstrated the largest number of qubit entanglements. For large integration, however, solid state quantum computers potential provide the best platforms. Examples for the two-state systems required for qubits include semiconductor quantum dots, superconducting device, the nuclear spins of donors in silicon, the electron spins of donors in SiGe heterostructures, and the use of surface acoustic waves with low dimensional structures in GaAs heterostructures.

Quantum computing is presently at an immature experimental stage but if many of the problems can be overcome, the potential power of the technique will allow many numerically intensive calculations to be completed that are presently impossible with classical computational machines.

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Digital systems

Martin Plonus, in Electronics and Communications for Scientists and Engineers (Second Edition), 2020

9.8 Quantum computers

Digital computers are based on digital logic. In recent years, significant progress is being made in creating a new kind of computer: the quantum computer. In many ways, a quantum computer is radically different from a digital computer. In particular, it is based on quantum logic gates. In principle, a quantum computer can perform every task that a digital computer can. However, there are some specialized tasks for which a quantum computer is much, much faster than a digital computer. Two such tasks are factoring of large numbers and data base searches.

Consider first the case of factoring large numbers. Suppose you have two prime numbers, X and Y. It is easy and fast to determine the product of those numbers: Q = XY. However, if someone hands you a large number, and tells you that it is a product of two prime numbers, it is very difficult and time consuming to determine the two factors. For example, if the number to be factored has 100 digits, even the fastest supercomputer will take more than a year to find the factors. Furthermore, the amount of time it takes to find the factors grows exponentially with the number of digits. Thus, if the number of digits (N) is much larger than 100, it is virtually impossible to find the factors on a reasonable time scale. On the other hand, a quantum computer consisting of only a few hundred gates can find the factors of a 100 digit number in a few seconds. For larger numbers, the amount of time needed for a quantum computer to find the factors grows only polynomially with N.

The fact that it is difficult to factor a large number, even with a supercomputer, is at the heart of the so-called RSA cryptography scheme, which is routinely used by many companies, including banks, to ensure the safety and security of financial transactions. This cryptography scheme would become obsolete if it were possible to realize a quantum computer with a few hundred gates.

Consider next the case of finding an item in a random database containing N elements. An example of such a database is a list of phone numbers in a phone book. If you want to find the name of a person for a given phone number, you have to make, on average, N/2 queries. On the other hand, a quantum computer can find the name by making only √N queries.

In addition to these two applications, quantum computers are also essential for efficient simulations of symptoms that behave quantum mechanically. In fact, the concept of a quantum computer was first put forth by Richard Feynman in order to address this challenge. Since the technology of quantum computing is in its infancy, it is plausible that other applications of this technology will also emerge soon. Furthermore, the technologies that are required for making a quantum computer can also be useful for secure communication as well as precision metrology. For these reasons, it is important to understand the elements of quantum computing.

Obviously, a quantum computer is based on the laws of quantum mechanics, which can be summarized as follows. Consider, for example, an electron in the presence of a magnetic field. It is well known from the laws of electromagnetism: a circulating current behaves like a magnet. An electron is always rotating, thus behaving like a magnet. In the presence of an external magnetic field, it rotates in either clockwise or counterclockwise direction with respect to the direction of the magnetic field. We denote as | 0⟩ the state for which the motion is counter-clockwise, and as | 1⟩ the state for which the motion is clockwise. Due to magnetic dipole interaction, the energy of the electron in state | 1⟩ is more than its energy in state | 0⟩.70 Let us denote by E0 the energy of state | 0⟩ and by E1 the energy of state | 1⟩. In the classical world, the electron can be only in state | 0⟩ or | 1⟩. However, according to the laws of quantum mechanics, the electron can be simultaneously in a superposition of state | 0⟩ and state | 1⟩.

Explicitly, we can write the quantum state of the electron as follows:

(9.46)∣Ψ⟩=C0|0〉+C1|1〉

This is also illustrated in Fig. 9.40.

Fig. 9.40. Schematic illustration of an electron in a magnetic field.

In general, the coefficients C0 and C1 are complex numbers, and | C0 |2 + | C1 |2 = 1. The physical meaning of these coefficients is as follows. Let us assume that we have a technique for determining whether the electron is in state | 0⟩ or state | 1⟩. If we use this technique, we will find the electron only in state | 0⟩ or in state | 1⟩. If we create this quantum state N times, then, we will find the electron in state | 0⟩ for | C0 |2N trials, and in state | 1⟩ for | C1 |2N trials. Thus, | C0 |2 is the probability of finding the electron in state | 0⟩, and | C1 |2 is the probability of finding the electron in state | 1⟩.

In the language of quantum mechanics, it is customary to employ a matrix notation where each state is represented by a column matrix

(9.47)|0〉≡[10];|1〉≡[01]

Using this notation, the general quantum state can be expressed as:

(9.48)|Ψ〉=C0|0〉+C1|1〉=[C0C1]

The quantum state, | Ψ⟩, obeys the so-called Schroedinger Equation, which can be expressed as follows:

(9.49)iℏ∂|Ψ〉∂t=H|Ψ〉

where H is the Hamiltonian. The Hamiltonian is simply another name for the energy of the system. For example, when considering the quantum mechanical description of an electron in free space, in the absence of any fields, the energy, and therefor the Hamiltonian, is simply the kinetic energy: p2/(2m), where p is the momentum and m is mass of the electron.

For the electron in a magnetic field, with two energy states, as being considered here, the Hamiltonian, in the matrix form, can be expressed as:

(9.50)H=[H11H12H21H22]

and ħ is the Planck's constant, h, divided by 2π. The values of the elements of the Hamiltonian depends on the details of the system. As an example, consider a situation where, in addition to the static magnetic field in the ˆz direction, as shown in Fig. 9.40, we apply another oscillating magnetic field, in the ˆx direction:

(9.51)B=Boˆz+B1ˆxB1=B10cos

The oscillation frequency, ω, is chosen to satisfy the following condition:

(9.52)ω=E1–E2/ħ

Eq. (9.53) indicates that the characteristic energy associated with this oscillation, given by ℏω, matches the energy difference between states | 1⟩ and | 0⟩. Under this condition, the Hamiltonian can be expressed as:

(9.53)H=ℏ20ΩΩ0

where the parameter Ω is called the Rabi frequency, and its value is proportional to the magnitude of the magnetic field in the xˆ direction:

(9.54)Ω=kB10

where k is a constant, which depends on inherent properties of the electron.

The equation of motion dictated by Eq. (9.49) can now be expressed as:

(9.55)iℏ∂C0t∂t∂C1t∂t=ℏ20ΩΩ0C0tC1t

Consider a situation where, at t = 0, the electron is in state | 0⟩ only:

(9.56)C00=1;C10=0

For this initial condition, it is easy to show that

(9.57)C0t=cosΩ2tC1t=−isinΩ2t

Thus, the probabilities of finding the electron in states | 0⟩ and | 1⟩, given by P0(t) = | C0(t)|2 and P1(t) = | C1(t)|2, respectively, can be expressed as:

(9.58)P0t=cos2Ω2t=121+cosΩtP1t=sin2Ω2t=121−cosΩt

This is illustrated in Fig. 9.41.

Fig. 9.41. Illustration of the Rabi oscillation.

This behavior is known as the Rabi oscillation. Note that the sum of P0(t) and P1(t) is always unity.

Consider now a situation where the oscillating field in the xˆ direction is applied for a duration, T, and then turned off. The duration of the pulse is characterized by the parameter ξ = ΩT. For example, if ξ = π, it is called a π pulse, and when ξ = π/2, it is called a π/2 pulse. Thus, if the electron starts in state | 0⟩, then, after a π-pulse, it is in state | 1⟩. On the other hand, the quantum state after a π/2 pulse is as follows

(9.59)Ψπ2=120−i1]

This means that the quantum state is in a superposition of states | 0⟩ and states | 1⟩, with equal probability of being in state | 0⟩ and state | 1⟩. Specifically, we now have C0=1/2, and C1=−i/2, so that the probability of finding the electron in state | 0⟩ is given by | C0 |2 = 1/2, and the probability of finding it in state | 1⟩ is given by | C1 |2 = 1/2. The fact that the amplitudes (i.e., C0 and C1) can be complex is an inherent feature of quantum mechanics. However, any quantity that can be measured (such as the probability of finding the electron in one state or the other) is always real.

An electron that follows the equation of motion given by Eq. (9.55) above is called a "quantum bit," or a "qubit" in short. This is the quantum mechanical counterpart of a classical bit in conventional digital logic. However, a key distinction between a classical bit and a quantum bit is now obvious: a classical bit can be in either state | 0⟩ or state | 1⟩, while a quantum bit can be in a superposition of states | 0⟩ and | 1⟩.

Another important and crucial aspect of quantum mechanism that underlies the power of quantum computing is the phenomenon of entanglement.71 To illustrate this, consider two different electrons, A and B. Let us assume that these are placed in magnetic fields, as described above, but are kept "far" away from each other, so they do not affect each other. The quantum states of these electrons are distinct, and can be expressed as:

(9.60)ΨA=α0A+β1AΨB=γ0B+δ1B

where α, β, γ, and δ are arbitrary parameters, obeying the conditions that | α |2 + | β |2 = 1 and | γ |2 + | δ |2 = 1. The combined quantum state for both particles can be expressed as:

(9.61)ΨAB=ΨA⊗ΨB=α0A+β1A⊗γ0B+δ1B=αγ0A0B+αδ0A1B+βγ1A0B+βδ1A1B

Here, the symbol ⊗ indicates an outer product, as illustrated in Eq. (9.61).

The quantum state expressed in Eq. (9.61) is called an "un-entangled" state, since it can be expressed as the outer product of distinct states for electron A and electron B. However, consider now a state of the following form

(9.62)ǀΨ~⟩AB=ϵ0A0B+η1A1B

It is easy to see that this state cannot be expressed as the outer product of distinct states for electron A and electron B. Such a state is called an "entangled" state.

To appreciate the implication of an entangled state of the form shown in Eq. (9.62), consider a situation where the two electrons are far apart, e.g., are in Chicago [A] and the other in London [B]. The quantum state, | 0⟩A | 0⟩B, represents a reality where both electrons are in state | 0⟩. Similarly, the quantum state | 1⟩A | 1⟩B represents a reality where both electrons are in state | 1⟩. Suppose now a person located in Chicago—let us call her Alice—measures the state of her electron to be in state | 0⟩. This is only possible if the joint system were in state | 0⟩A | 0⟩B. Thus, if the person located in London—let us call him Bob—measures the state of his electron, he will find it to be in state | 0⟩ as well.

To see how to produce an entangled state, recall first the concept of logic gate operations in the world of conventional digital logic. For example, the NOT operation converts a 0 to 1 and vice versa. Similar logic gate operations can be constructed for quantum bits. Of particular importance for quantum logic is the so called controlled-NOT gate, which is also called CNOT gate. To see how it works, consider a situation where you have two qubits, A and B. Let us assume that, before the CNOT operation, the quantum states of the two particles are as follows:

(9.63)ΨA=α0A+β1AΨB=0B

so that the combined state (which is un-entangled at this point) can be expressed as

(9.64)ΨAB=ΨA⊗ΨB=α0A0B+β1A0B

We now apply the CNOT{A, B} gate operation which flips the state of B from | 0⟩ to | 1⟩ or | 1⟩ to | 0⟩ only if the state of A is | 1⟩. Thus

(Note, the state of A is a superposition of | 0⟩ and | 1⟩, as indicated by the first line of Eq. 9.63)

(9.65)CNOTAB→ΨAB⇒Ψ~AB=α0A0B+β1A1B

that is, application of the CNOT{A, B} operation on the un-entangled state | Ψ⟩AB produces the entangled state Ψ~AB.

It is not difficult to understand how to realize a CNOT gate. Consider a situation where the two electrons, A and B, are "close" to each other, in the presence of a constant magnetic field in the zˆ-direction: B⃑=B0zˆwhere B0 is produced by an external system. Now, in addition to this field, electron B will see the additional magnetic field produced by electron A (Recall that the electron is spinning, either in the clockwise or the counter-clockwise direction, and a spinning electron acts as a magnet, producing a magnetic field). Thus when electron A is in state | 0⟩, it produces an additional component of the magnetic field, at the location of electron B, which reduces the net magnetic field seen by B. Similarly, when A is in state | 1⟩, the net magnetic field seen by B is higher. This is illustrated in Fig. 9.42:

Fig. 9.42. Effect of the state of A on the magnetic field seen by B.

Recall also that the energy difference (E1 – E0), for an electron is proportional to the field in the zˆ-direction. Thus, for B1, (E1 – E0) is lower when A is in state | 0⟩ and higher when A is in state | 1⟩. Recall also that a Rabi oscillation between states | 0⟩ and | 1⟩ takes place only when we apply a transverse oscillating magnetic field in the xˆ - direction, at a frequency ω so that ℏω = (E1 – E0). Thus if we apply an oscillating magnetic field with a frequency ω = (E1 – E0)/ℏ corresponding to the field B0 + (as defined in Fig. 9.42), the Rabi oscillation will occur for B only if A is in state | 1⟩, and the state of B will remain unaffected if A is in state | 0⟩. If we choose the duration of the Rabi oscillation to be a π-pulse, the application of this oscillating field will realize a CNOT{A, B} gate.

We recall that the NAND gate is the universal logic gate for conventional digital logic. Similarly it can be shown that a universal quantum computer can be realized, for a system with N qubits if we can perform the following operations:

Single Qubit Operation: The ability to produce Rabi oscillation between | 0⟩ and | 1⟩ for each qubit (SQO)

General Two-Qubit Operation: The ability to carry out a CNOT operation among any two qubits (CNOT).

This is illustrated schematically in Fig. 9.43. Here, we have shown N quantum bits, placed in a linear array. The arrows labeled as SQOs represent the fact that it is possible to perform single qubit operations on each qubit, without affecting the quantum state of any other qubit. The arrows labeled as CNOT{i, j} represent the fact that it is possible to carry out a Controlled-NOT gate operation between the i-th qubit and the j-th qubit.

Fig. 9.43. A universal quantum computer.

In practice it can be shown that if we are able to carry out a CNOT operation between two neighboring qubits, many such CNOT operations can be cascaded to realize a CNOT operation between any two qubits. As such, a simpler quantum computer can be implemented in the manner shown in Fig. 9.44. Here, the CNOT gate operations can be carried out between nearest neighbor quantum bits only.

Fig. 9.44. A simplified quantum computer.

Recall that, in conventional digital logic, a register with N bits can store one of 2N numbers, ranging in values from 0 to (2N – 1). For example, for N = 3, these numbers are

(9.66)0=0001=0012=0103=0114=1005=1016=1107=111

We now note that the individual quantum states of an N-qubit quantum computer can be expressed in a similar fashion:

(9.67)ϕ0≡010203...0Nϕ1≡010203...1Nϕ2≡010203…1N−10N⋮ϕ2N≡111213...1N

Using these individual states, we can write the general quantum state of an N-bit quantum computer as:

(9.68)Ψ=∑j=02N−1cjϕj

with the constraint that

(9.69)∑j=02N−1cj2=1

The equation of motion for the general quantum state in Eq. (9.68) can be written as:

(9.70)iℏ∂Ψ∂t=HΨ

Just as before, we can represent each individual quantum state as a column matrix (1 × 2N column)

ϕ0=100⋮0;ϕ1=010⋮0;…ϕ2N−1=000⋮1

so that the complete state for the quantum computer can be expressed as:

Ψ=c0c1c2⋮c2N−1

Similarly, the Hamiltonian, H, which represents the single qubit operation and the CNOT gates, can, in general, be represented by a square matrix with 2N rows and 2N columns.

A system that realizes all the elements of the Hamiltonian (some of which can have 0 values) in a controllable manner is called a quantum computer.

For practical reasons, it is very difficult to make a quantum computer with a large number of qubits, So far, the maximum number of qubits that have been realized in a quantum computer is ~ 15. Hopefully, in the not-to-distant future, it will be possible to realize quantum computers with a very large number of qubits.

In the introduction to this chapter, it was mentioned that the most important application of a quantum computer is the factoring of a large number that is the product of two prime numbers (e.g., 15 = 3 × 5). It is natural to expect that an example of such a factoring process using a quantum computer would be presented here. However, it turns out that this process is quite elaborate and complicated. First, it requires the exposition of some theorems in number theory. Second, it requires a discussion of how Fourier Transforms can be done with a quantum computer. Third, the result is probabilistic, and multiple trials are needed to get the answers. A reader interested in learning about the exact steps can read, for example, the excellent review article by Ekert and Jozsa ("Quantum computation and Shor's factoring algorithm," Artur Ekert and Richard Jozsa, Review of Modern Physics, Reviews of Modern Physics, Vol. 68, No. 3, July 1996, pp. 733–753) which can be found at https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.68.733.

Example 9.27

Consider the expression shown in Eq. (9.55). For C0(0) = 0 and C1(0) = 1, find the solution for C0(t) and C1(t).

Solution

dC0tdt=−iΩ2C1t;dC1tdt=−iΩ2C0t➔d2C1tdt2=−Ω24C1t

The general solution of this equation is:

C1t=AcosΩ2t+BsinΩ2t

We then have:

C0t=i2ΩdC1tdt=i2Ω−Ω2AsinΩ2t+Ω2BcosΩ2t

We know:

C00=0➔B=0

C10=1➔A=1

We thus get:

C1t=cosΩ2tC0t=−isinΩ2t

Example 9.28

Consider the following quantum states. Determine the values of the unspecified parameter 'N' for each state:

(i)

Ψ=1N20+31

(ii)

Ψ=1N120+131

(iii)

Ψ=150N0+71

(iv)

Ψ=1N−3i0+41

Solutions

In general, | Ψ⟩ = C0 | 0⟩ + C1 | 1⟩, with the condition that | C0 |2 + | C1 |2 = 1. Using this rule, we get

(i)

1N4+9=1➔N=13

(ii)

1N14+19=1➔N=1336

(iii)

150N2+49=1➔N=eiϕforanyvalue ofϕ.

For example, if ϕ = 0, then N = 1; if ϕ = π, then N = − 1; if ϕ=π2, then N = 2; and so on. In all cases, | N |2 = 1.

(iv)

1N9+16=1➔N=25

Example 9.29

Determine the resulting state when CNOT{A, B} is applied to the following states:

(i)

Ψ〉=350〉A0〉B+451〉A1〉B

(ii)

Ψ〉=121〉A0〉B+i21〉A1〉B

Solution

(i)

CNOTABΨ〉=350〉A0〉B+451〉A0〉B

(ii)

CNOTABΨ〉=121〉A1〉B+i21〉A0〉B

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Quantum Computation and Chaos

G. Casati, G. Benenti, in Encyclopedia of Condensed Matter Physics, 2005

Quantum Noise

Any practical implementation of a quantum computer will have to face errors, due to the inevitable coupling of quantum processors to the surrounding environment or to device imperfections. The first kind of error is known as decoherence and is a threat to the actual implementation of any quantum computation. More generally, decoherence theory has a fundamental interest beyond quantum information science, since it provides explanations for the emergence of classicality in a world governed by the laws of quantum mechanics. The core of the problem is the superposition principle, according to which any superposition of quantum states is an acceptable quantum state. This entails consequences that are absurd according to classical intuition, like the superposition of "cat alive" and "cat dead" that is considered in the Schrödinger's cat paradox. The interaction with the environment can destroy the coherence between the states appearing in a superposition (for instance, the "cat alive" and "cat dead" states). Therefore, decoherence invalidates the quantum superposition principle, which is at the heart of the power of quantum algorithms.

The presence of device imperfections, although not leading to any decoherence, also hinders the implementation of any quantum computational task. Imperfection effects can be modeled as follows: the quantum computer is seen as a lattice of interacting spins (qubits) where, due to the unavoidable presence of imperfections, the spacing between the up and down states and the couplings between the qubits are both random. The Hamiltonian of this model reads, for a linear array of n qubits, as follows:

[23]HS=∑i(Δ0+δi)σiz+∑i

where the σi are the Pauli matrices for the qubit i, and Δ0 is the average level spacing for one qubit. The second sum in eqn [23] runs over nearest-neighbor qubit pairs and δi, Jij are randomly and uniformly distributed in the intervals [−δ/2, δ/2] and [−J, J], respectively. It is assumed that the phase accumulation given by Δ0 can be eliminated by standard spin echo techniques, while the other terms give unwanted phase rotations and qubit couplings. In Figure 3, the limits to quantum computation due to hardware imperfections in a concrete example are illustrated, the quantum algorithm simulating the sawtooth map. The fidelity of quantum computation, defined by f(t)=|〈ψε(t)|ψ0(t)〉|2 is plotted, where |ψε(t)〉 is the actual quantum wave function in the presence of imperfections and |ψ0(t)〉 is the quantum state for a perfect computation. One can show that the fidelity drops with time according to the law f(t)≈exp(−At2), as expected, for small ɛ, from perturbation theory. The constant A∝nq(εng)2, with ng=3n2+n number of quantum gates per map iteration. Therefore, a reliable quantum computation is possible up to a timescale tf∝ε−1ng−1nq−1/2. A longer quantum computation would necessarily require the implementation of quantum error correcting codes.

Figure 3. Fidelity as a function of t, for the sawtooth map with parameter values as in Figure 1, n = 9 qubits, J = δ (red lines) and J = 0 (blue lines). From top to bottom: imperfection strength ε = δτg = 10−5, 3×10−5, 10−4, 3×10−4, 10−3. Here τg denotes the time interval between consecutive quantum gates.

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Quantum chemistry on a quantum computer

Guido Fano, S.M. Blinder, in Mathematical Physics in Theoretical Chemistry, 2019

7 Time-evolution of a quantum system

The principal goal of the quantum-computer program we are outlining is to find accurate values of energy eigenvalues, most often just the ground-state energy. The method exploits the fact that the time-evolution of a quantum system exhibits its eigenvalues, in the form of a Fourier superposition of its eigenstates. Let us begin with the time-dependent Schrödinger equation for an arbitrary quantum state Ψ(x1, x2, …, xN, t), abbreviated Ψ(x, t):

(74)i∂Ψ(x,t)∂t=HΨ(x,t).

The formal solution, with an initial state Ψ(x, 0), can be written

(75)Ψ(x,t)=e−iHtΨ(x,0),

where the exponential, known as the evolution operator or propagator, is explicitly a representation of the power series

(76)e−iHt=∑n=0∞(−iHt)nn!=1+(−it)H+(−it)22!H2+(−it)33!H3+⋯.

The initial state Ψ(x, 0) is formally represented by a superposition of eigenstates of the Hamiltonian H:

(77)Ψ(x,0)=∑mcmψm(x)=c0ψ0(x)+c1ψ1(x)+c2ψ2(x)+⋯+continuum,

where Hψm(x) = Emψm(x) = ωmψm(x). It is useful to express the energies in frequency units Em=ℏωm=ωm. We label the ground state as m = 0. Applying the evolution operator, we find

(78)Ψ(x,t)=e−iHtΨ(x,0)=∑mcme−iωmtψm(x)=c0e−iω0tψ0(x)+c1e−iω1tψ1(x)+⋯+continuum.

The scalar product of Ψ(x, t) with Ψ(x, 0) is an instance of an autocorrelation function:

The Fourier transform of the autocorrelation function exhibits the spectrum of eigenvalues

(80)G(ω)=12π∑m|cm|2δ(ω−ωm)+continuum.

In practice, the peaks of G(ω) will be finite, because of approximations and inaccuracies, something like the spectrum shown in Fig. 17.

Fig. 17. Simulation of computed spectrum in quantum computation.

The objective pursued in this chapter is computation of the quantized energy levels of a molecular system. This will be realized by simulation of the dynamics of the system, generated by the time-evolution operator e−iHt. To do this on a quantum computer, we must construct a set of one- and two-qubit gates that implement this exponential transformation on a set of qubits. Approximations to the energy eigenvalues then appear in the phases of qubit states, which are determined using the phase-estimation algorithm.

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Computational Error and Complexity in Science and Engineering

V. Lakshmikantham, S.K. Sen, in Mathematics in Science and Engineering, 2005

3.10 Quantum complexity

Recently to analyze the computational power of quantum computers (no commercial quantum computers are so far existing), the quantum complexity is studied. R. Feynman (1982) observed that the conventional computers based on silicon technology could not efficiently simulate the quantum systems. He felt that if a computer could be built based on quantum mechanics then it might be able to perform the task more efficiently. Such a theoretical computational model was developed by D. Deutch (1985). Two quantumalgorithms (Shor 1997; Grover 1996) received significant attention. One algorithm was for factoring an integer in a polynomial time on a quantum machine while the other for searching a database of n elements in O(√n) operations/time.

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Quantum Mechanics Fundamentals

Ivan Djordjevic, in Quantum Information Processing and Quantum Error Correction, 2012

2.4.2 Pauli Operators

The basic unit of information in a quantum computer is known as a quantum bit or qubit. The corresponding state space is two-dimensional with the following basis: {|0〉 = [1 0]T,|1〉 = [0 1]T}. The arbitrary state ket can be written in terms of basis kets as follows:

(2.47)|ψ〉=α|0〉+β|1〉,

with α and β satisfying the following normalization condition: |α|2+|β|2=1. The probability that a qubit upon measurement is in a state |0〉 is determined by |〈0|ψ〉|2 = |α|2.

The Pauli operators X, Y, Z (very often denoted as σx, σy, and σz or σ1, σ2, and σ3) correspond to the measurement of the spin along the x-, y-, and z-axes respectively. Their actions on basis states are given by

(2.48)X|0〉=|1〉,X|1〉=|0〉Y|0〉=−j|1〉,Y|1〉=j|0〉Z|0〉=|0〉,Z|1〉=−|1〉.

It is clear that basis states are eigenkets of Z. We have shown earlier that an operator Ξ can be represented in matrix form in {|a(k)〉} basis with matrix elements given by Ξij = 〈a(i)|Ξ|a(j)〉. Based on the action of the Pauli X operator in (2.48), it can be represented in matrix form as

(2.49)X=˙(〈0|X|0〉〈0|X|1〉〈1|X|0〉〈1|X|1〉)=(〈0|1〉〈0|0〉〈1|1〉〈1|0〉)=(0110).

In similar fashion, the Pauli X and Y operators can be represented as

(2.50)Y=˙(0−jj0),Z=˙(100−1).

Since any operator Ξ can be written in terms of outer product as Ξ=∑n=1∞∑m=1∞Ξmn|a(m)〉〈a(n)|, the Pauli X operator can be written as

(2.51)X=(0X01X100)=(0110)=X01|0〉〈1|+X10|1〉〈0|=|0〉〈1|+|1〉〈0|.

(2.52)Z=|0〉〈0|−|1〉〈1|.

The projection operators corresponding to measurements of 1 and −1 can be defined as

(2.53)P0=|0〉〈0|,P1=|1〉〈1|.

The projector P0 (P1) performs projection of an arbitrary state to the |0〉 (|1〉) state:

(2.54)|ψ〉=α|0〉+β|1〉P0|ψ〉=(|0〉〈0|)|ψ〉=α|0〉,P1|ψ〉=(|1〉〈1|)|ψ〉=β|1〉.

The sum of projection operators clearly results in the identity operator as shown below:

(2.55)P0+P1=|0〉〈0|+|1〉〈1|=(10)(10)+(01)(01)=(1000)+(0001)=(1001)=I.

|αk|2=|〈ξ(k)|ψ〉|2=〈ξ(k)|ψ〉〈ξ(k)|ψ〉∗=〈ξ(k)|ψ〉〈ψ|ξ(k)〉=〈ψ|ξ(k)〉〈ξ(k)|ψ〉=〈ψ|Pk|ψ〉.

We have also shown that the final state of the system of the measurement will be Pk|ψ〉/〈ψ|Pk|ψ〉. For a two-dimensional system, if the measurement result +1 is obtained the result after the measurement will be

(2.56)|ψ′〉=1〈ψ|P0|ψ〉P0|ψ〉=|ψ〉=α|0〉+β|1〉α|α||0〉,

and if measurement result −1 is obtained the corresponding state will be

(2.57)|ψ′〉=1〈ψ|P1|ψ〉P1|ψ〉=β|β||1〉.

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A Formalism for Quantum Computing and a Satisfiability Problem

C. Bautista, ... A.F.K. Zehe, in Recent Advances in Multidisciplinary Applied Physics, 2005

INTRODUCTION

There is little doubt, that nanotechnology will create the building blocks for practical quantum computers, consisting most probably in a set of coupled solid-state quantum systems, carefully put into place. On the other hand, efficient algorithms are required by the quantum information technology in order to meet the high expectations on speed and security.

Since the discovery of a polynomial quantum algorithm for factoring integers by Shor [1], quantum computation has drawn overwhelming attention from the science community, in particular from physicists and mathematicians interested in computer science.

Chuang et al. [2] proved the technological feasibility of building a quantum computer that can factorize the number 15. However a functional real quantum computer remains to be built, to the best of our knowledge. It is a general believe that it is just a technological problem that will be overcome [3,4] some day.

A quantum computer is some kind of probabilistic machine with the additional feature of allowing superpositions of states which lead to a massive parallel processing and, as a consequence, to an amazing speedup of calculation. Excellent introductions and enjoyable readings, to quantum computation, are found in [11, 12].

Despite that at this moment, the most important problem in quantum computation is its physical realization, the design of quantum algorithms still is a very important issue. According with Shor [5] it may seem that computer scientists have to be accustomed to the very particular specifications of quantum computing, because most of these are far away from our daily experience (quantum mechanics).

The main objective of this paper is to show how quantum computation could deal with classical subjects of computing theory, like the SAT problems.

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