Problems and Solutions in Quantum Computing and Quantum Information

The following sections are included:

• Preface

• Contents

• Notation

A single qubit is a two-state system, such as a two-level atom. The states (kets) |h〉 and |v〉 of the horizontal and vertical polarization of a photon can also be considered as a two-state system. Another example is the relative phase and intensity of a single photon in two arms of an interferometer. The underlying Hilbert space for the qubit is C²…

Let ℋ₁ and ℋ² be two Hilbert spaces and ℋ be a third Hilbert space defined in terms of ℋ₁ and ℋ₂ with the following specifications. For each pair of vectors f₁, f₂ in ℋ₁, ℋ₂, respectively, there is a vector in ℋ denoted by f₁ ⊗ f₂ such that

where 〈f₁|g₁〉 is the scalar product in the Hilbert space ℋ₁…

For finite-dimensional quantum systems finding the norm, eigenvalues, eigenvectors, Schmidt rank, inverse (if it exists) of square matrices are important. Let A be an n×n matrix over C…

A density operator ρ or density matrix is a positive semidefinite operator on a Hilbert space with unit trace. An operator is positive semidefinite if it is hermitian and none of its (necessarily real) eigenvalues are less than zero. The state of a quantum-mechanical system is characterized by a density operator ρ with trρ = 1…

Let be the finite dimensional Hilbert space Cⁿ with an orthonormal basis {|ø_j〉 :j = 1,2,…, n }. Let A be a linear operator (n × n matrix) acting in this Hilbert space. Then the trace of A is defined as

…

Quantum gates are realised as unitary operators. Let denote a Hilbert space. A linear operator U in a Hilbert space is unitary if

. In other words, U is unitary if and only if U is invertible and U⁻¹ = U∗…

For any density operator

the von Neumann entropy is defined as or equivalently with 0log0 = 0. Thus the von Neumann entropy is equal to the Shannon entropy of the eigenvalues…

In quantum measurement models we consider what kind of measurements can be made on quantum systems as well as how to determine the probability that a measurement yields a given result. The effect that measurement has on the state of a quantum system is also important…

Entanglement is the characteristic trait of quantum mechanics which enforces its entire departure from classical lines of thought. Let and be two finite-dimensional Hilbert spaces…

Bell's theorem states that, according to quantum mechanics, the value of a certain combination of correlations for experiments of two distant systems can be higher than the highest value allowed by any local-realistic theory of the type proposed by Einstein, Podolsky and Rosen, in which local properties of a system determine the result of any experiment on that system. The most discussed Bell inequality, the Clauser-Horne-Shimony- Holt (CHSH) inequality states that in any local-realistic theory the absolute value of a combination of four correlations is bounded by 2. Consider a system with two distant particles i and j…

Teleportation is the transmission of quantum information using a classical channel and entanglement. It demonstrates the use of entanglement as a communication resource. The simplest case is to consider the teleportation of a single qubit using two bits of classical communication and one entangled pair (EPR-pair). Quantum teleportation is the disembodied transport of an unkown quantum state from one place to another. The key idea is that two distant operators, Alice at a sending station and Bob at a receiving terminal, share an entangled quantum bipartite state and exploit its nonlocal character as a quantum resource. The resource state can be the singlet state of a pair of spin-½ particles

Particle 1 is given to Alice and particle 2 is given to Bob. Alice intends to transport an unkown state of a third spin-1 particle to Bob. She performs a complete projective measurement on the joint system consisting of particle 1 and 3 and then conveys its outcome to Bob via a classical communication channel. As a consequence of Alice's measurement, the total-spin state of the three-particle system collapses. Owing to the entanglement, this involves a breakdown of the spin-| state of Bob's particle 2. Nevertheless, Bob makes use of the information transmitted classically by Alice to transform his reduced state into an output that is an accurate replica of the original unknown input…

Cloning is the duplication of information. Cloning is necessarily a physical process. Obviously we have to find unitary transformations for the cloning process. The no cloning theorem is a result of quantum mechanics which forbids the creation of identical copies of an arbitrary unkown quantum state. However, approximate copies still have many uses in quantum computing. Due to the superposition principle of quantum mechanics it it not possible in general to clone an arbitrary quantum state. As a consequence any succesfull attempt to clone a state will destroy the original state in the process…

An algorithm is a precise description of how to realize a given objective, for example solving a computational problem. We distinguish between classical and quantum algorithms where quantum physical resources are used. Deutsch's problem leads to the simplest quantum algorithm. Consider the Boolean functions f that map {0, 1} to { 0, 1 }. There are exactly four such functions: two constant functions (f(0) = f ( 1 ) = 0 and f(0) = f ( 1 ) = 1) and two balanced functions (f(0) = 0, f(1) = 1 and f(0) = 1,f(1) = 0)…

In classical communication theory where bits are communicated, the only possible type of error that can occur is a bit flip. In the quantum case any rotation or phase change in the Hilbert space of the quantum state is an error, i.e., there are an infinite number of different errors that could occur just for a single qubit. Fortunately the measurement process involves the projection of the quantum state into a compatible subspace. Thus measurement to determine the occurrence of an error reduces the error to one compatible with the measurement. Suppose some data is contained in the state |x〉, and the environment is described by |E〉…

Cryptography usually involves a key or keys to be used in encryption and decryption algorithms. Classical cryptography generally relies on maps that are perceived to be very difficult to invert with incomplete information. One popular algorithm due to Rivest, Shamir and Adelman is the RSA algorithm

…

Besides qubit-based quantum computing and quantum algorithms, quantum information over continuous variables is also applied and used in fields such as quantum teleportation and quantum cryptography. For continuous systems Bose operators play the central role…

Quantum coherent states are the closest quantum-mechanical analogue to a classical particle oscillating in a harmonic potential. Coherent states are minimum uncertainty states. Quantum computation circuits with coherent states as the logical qubits can be constructed using simple linear networks, conditional measurements and coherent superposition resource states. Coherent states are very sensitive to their environment. The output of a single mode stabilised laser can be described by a coherent state |β〉, where β is a complex number which determines the average field amplitude. Harmonic oscillator coherent states can be defined in three different equivalent ways…

Coherent states are not the most general kind of Gaussian wave packet. They are also not the most general kind of minimum-uncertainty wave packets, since the minimum uncertainty wave packet satisfies ΔqΔp = ħ/2 which only constrains the product of the dispersions Δq and Δp, whereas for coherent states we have that (Δq)² = ħ/(2ω) and (Δp)² = ħω/2. For squeezed states we do not have this restriction. Unlike a coherent state, an initial squeezed state does not remain a minimum-uncertainty state in the course of time under the harmonic oscillator evolution. Instead, the product ΔqΔp oscillates at twice the harmonic oscillator frequency between a maximum value and a minimum value. Squeezed states possess the property that one quadrature phase has reduced fluctuations compared to the ordinary vacuum. Squeezed states of the electromagnetic field are generated by degenerate parametric down conversion in an optical cavity…

In the original paper of Einstein, Podoisky and Rosen the spin version of entanglement was not used, rather they considered measurement of position and momentum observables for two particles in one-dimensional motion. They considered the entangled state

where the first component in the tensor product refers to particle 1 and the second to particle 2…

Quantum continuous variables provide a new approach to quantum information processing and quantum communication. They describe highly excited quantum systems such as multi-photon fields of light. Continuous variables offer additional advantages over the single-photon system. They involve the use of highly efficient telecommunication photodiodes. The coherent sources of continuous entanglement are also orders of magnitude more efficient than the spontaneous sources of discrete entanglement. Teleportation schemes can be demonstrated involving bright light sources…

Swapping and cloning cannot only be studied for finite-dimensional systems but also for continuous variables. We can therefore investigate whether coherent states can be swapped or cloned…

Most experimental realizations of quantum logic gates (Hadamard gate, quantum phase gate, controlled-NOT gate) involve several qubits and number states (Fock states). A Hamilton operator Ĥ must describe the interaction. Thus in quantum computing we are faced with two problems. One is to determine the Hamilton operator H for the system such that the time-evolution exp(−iĤt/ħ) represents the execution of the computation. The other one is to build the hardware described by this Hamilton operator…

The following sections are included:

• Bibliography

• Index