In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, i.e. unitary operators acting on pure states (qubits or quregisters) of a convenient (n-fold tensor product) Hilbert space ⊗nC2 (where C2 is the bidimensional complex Hilbert space). Consequently, quantum gates represent reversible time evolution of the system. According to [1] however, such a representation is unduly restrictive insomuch it does not encompass open systems where coupling with environment and measurement processes may occur. In this case, time evolution is no longer reversible. In [1] the authors formulate a more general model of quantum computational processes, where pure states and unitary operators are replaced by density operators and quantum operations, respectively. Let H be a Hilbert space, L(H) the vector space of all linear operators of H and D(H) the set of density operators in H. A quantum operation [2] is a linear and completely positive operator O: L(H1) → L(H2). By [2], for every quantum operation O: L(H1) → L(H2), there exists a set of operators {Ai} such that iA iAi = I and ∀ρ ∈ D(H1): E(ρ) = iAiρA
In the usual representation of quantum computational processes, a quantum circuit is identified with an appropriate composition of quantum gates, i.e. unitary operators acting on pure states (qubits or quregisters) of a convenient (n-fold tensor product) Hilbert space ⊗nC2 (where C2 is the bidimensional complex Hilbert space). Consequently, quantum gates represent reversible time evolution of the system. According to [1] however, such a representation is unduly restrictive insomuch it does not encompass open systems where coupling with environment and measurement processes may occur. In this case, time evolution is no longer reversible. In [1] the authors formulate a more general model of quantum computational processes, where pure states and unitary operators are replaced by density operators and quantum operations, respectively. Let H be a Hilbert space, L(H) the vector space of all linear operators of H and D(H) the set of density operators in H. A quantum operation [2] is a linear and completely positive operator O: L(H1) → L(H2). By [2], for every quantum operation O: L(H1) → L(H2), there exists a set of operators {Ai} such that iA iAi = I and ∀ρ ∈ D(H1): E(ρ) = iAiρA
Abstract In quantum computational logics meaning of sentences are repre-sented by quantum information quantities. Any n-qubit sequence is identified with a unit vector of the tensor product of the complex Hilbert space ⊗nC2. Let us split the set of the vectors of its computational basis in the two sets: Bn1: = {i: ‖ i À= | x1,.., xn 〉 and xn = 1}; Bn0: = {i: ‖ i À= | x1,.., xn 〉 and xn = 0}. Any unit vector | ψ 〉 can be expressed in the form: | ψ 〉 = ∑i∈B0 ai ‖ iÀ +∑j∈B1 bj ‖ j À where by the Born rule,∑ i∈B0 | ai |2 + j∈B1 | bj |2 = 1 A key role in the definition of semantics is played by the probability-value of | ψ〉, defined as follows: p( | ψ〉): = ∑j∈B1 | bj |2 Moreover a function g: [0, 1]2 7 → [0, 1] is said quantum simulable iff ∃n ≥ 1, a unitary operator Ug: ⊗n+2C2 7 → ⊗n+2C2 and a quregister | χ 〉 ∈ ⊗nC2 such
This presentation provides a comprehensive overview of the foundational principles of quantum computation. It begins by contrasting classical and quantum information, highlighting how quantum bits (qubits) leverage superposition, entanglement, and interference to process information in ways that surpass classical limits. The document explains the four postulates of quantum mechanics: state space (qubits as unit vectors in a Hilbert space), time evolution (via unitary operators), measurement (probabilistic collapse of the state), and composite systems (tensor products of subsystems). Key concepts such as the Bloch sphere representation, quantum gates (e.g., Hadamard, CNOT, Pauli gates), and quantum circuits are discussed, illustrating how quantum operations manipulate qubit states. The presentation also covers quantum evolution, the gate model for computation, and the universality of certain gate sets, emphasizing the potential for quantum parallelism and the transformative impact of quantum computing on fields like cryptography and optimization.
In Section 1.3, we introduced the circuit model of (classical) computation. We restricted attention to reversible circuits since they can simulate any non-reversible circuit with modest overhead. This model can be generalized to a model of quantum circuits. In the quantum circuit model, we have logical qubits carried along ‘wires’, and quantum gates that act on the qubits. A quantum gate acting on n qubits has the input qubits carried to it by n wires, and n other wires carry the output qubits away from the gate. A quantum circuit is often illustrated schematically by a circuit diagram as shown in Figure 4.1. The wires are shown as horizontal lines, and we imagine the qubits propagating along the wires from left to right in time. The gates are shown as rectangular blocks. For convenience, we will restrict attention to unitary quantum gates (which are also reversible). Recall from Section 3.5.3 that non-unitary (non-reversible) quantum operations can be simulated by unitary (reversible) quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A circuit diagram describing a superoperator being implemented using a unitary operator is illustrated in Figure 4.2. In the example of Figure 4.1, the 4-qubit state |ψi⟩= |0⟩⊗ |0⟩⊗ |0⟩⊗ |0⟩ enters the circuit at the left (recall we often write this state as |ψi⟩ = |0⟩|0⟩|0⟩|0⟩ or |ψi⟩ = |0000⟩.) These qubits are processed by the gates U1, U2, U3, and U4. At the output of the circuit we have the collective (possibly entangled) 4-qubit state |ψf⟩. A measurement is then made of the resulting state. The measurement will often be a simple qubit-by-qubit measurement in the computational basis, but in some cases may be a more general measurement of the joint state. A measurement of a single qubit in the computational basis is denoted on a circuit diagram by a small triangle, as shown in Figure 4.1 (there are other symbols used in the literature, but we adopt this one). The triangle symbol will be modified for cases in which there is a need to indicate different types of measurements.R50
Usually models for quantum computations deal with unitary gates on pure states. In this paper we generalize the usual model. We consider a model of quantum computations in which the state is an operator of density matrix and the gates are quantum operations, not necessarily unitary. A mixed state (operator of density matrix) of n two-level quantum systems is considered as an element of 4n-dimensional operator Hilbert space. Unitary quantum gates and nonunitary quantum operations for n-qubit system are considered as generalized quantum gates acting on mixed state. In this paper we study universality for quantum computations by quantum operations on mixed states.
Quantum mechanics has proven its position as the primary theory for accurately describing microscopic domains. Numerous fields in physics utilize quantum mechanics to great effects, with one prominent application being quantum computing. Quantum computing extends the field of quantum mechanics to information theory, striving to perform computational tasks utilizing quantum phenomena with notably greater efficiency compared to classical alternatives. In this thesis, I will go through the basics required in the field of quantum computing. To understand quantum computing demands adapting a new type of matrix mechanical view. This involves considering Hilbert spaces, which are linear vector space having a well defined inner product. Hilbert spaces contain state vectors describing quantum states, as well as operators for altering and measuring these states. The dynamics of quantum systems are governed by three quantum postulates. These specify Hilbert spaces and state vectors as the fundamental components of quantum systems, Hermitian operators as providing the physical manifestation of these systems, and finally, the Schrödinger's equation to describe the time evolution of such systems. These notions are required in producing quantum computing. Quantum computing uses qubits as the computational basis, analogous to classical bits but with the added capability of quantum superposition. Qubits are represented as two-state vectors, and can be physically realized with any quantum-behaving binary-valued system, like the electron spins. Qubits can be combined into higher dimensional states with the use of tensor products, causing the system's Hilbert space to expand as 2^n with n being the number of qubits, allowing for exponential computational speedup. Computational logic can be constructed by acting on the qubits using unitary operators, known as quantum gates. The combinations of qubits and quantum gates can be visualized as quantum circuits, depicted using specialized circuit diagram notations. An essential part of quantum computing, and the phenomena enabling exponential computational speedup, is quantum entanglement. Entanglement arises in interacting qubits, intertwining them so that the individual states become correlated. Entanglement can be characterized by being unable to reduce a combined state into its pure constituents. The amount of entanglement in a system can be measured using von Neumann entropy, analogous to Shannon entropy in classical information theory. Entanglement serves various purposes in quantum computing and is a defining feature distinguishing it from the classical domain. In this thesis, the entanglement properties of a periodic four-qubit circuit with alternating quantum gates were simulated. The simulation was performed computationally, and measured entanglement entropy as a function of the amount of circuit periods. The results demonstrated different characteristics of entanglement depending on the parameters used for the quantum gates, with the entanglement either being periodic, non-occurring or chaotic in nature.
Let H1, H2 be complex Hilbert spaces, H be their Hilbert tensor product and let tr2 be the operator of taking partial trace, with respect to the space H2, of trace class operators in H. The operation tr2 maps states in H (=positive trace class operators in H with trace equal to 1) into states in H1. In this paper we give the full description of mappings that are linear right inverse to tr2. More precisely, we prove that any affine mapping F(W) of the convex set of states in H1 into the states in H that is right inverse to tr2 is given by W⟼W⊗D for some state D in H2. In addition we investigate a representation of the quantum mechanical state space by probability measures on the set of pure states and a representation—used in the theory of stochastic Schrödinger equations—by probability measures on the Hilbert space. We prove that there are no affine mappings from the state space of quantum mechanics into these spaces of probability measures.