This chapter discusses the main operation of tensor product, which is behind almost all quantum information manipulation. Tensor products are exploited in a similar manner to yield quantum operators to be used with quantum states of two or more quantum bits. The magnitude of the inversion depends on the state of the control q-bit. It can be said that the CNOT operator is a NOT operator dependent on a second q-bit. It can be constructed out of two chained H operators. The Toffoli operator can be seen as an extension of the CNOT port. It is considered universal and it is possible to build based on any other quantum operator. The chapter shows that if not handled efficiently, it would require a lot of memory space, which makes the design emulate a quantum processor quite expensive to produce.
This chapter discusses the main operation of tensor product, which is behind almost all quantum information manipulation. Tensor products are exploited in a similar manner to yield quantum operators to be used with quantum states of two or more quantum bits. The magnitude of the inversion depends on the state of the control q-bit. It can be said that the CNOT operator is a NOT operator dependent on a second q-bit. It can be constructed out of two chained H operators. The Toffoli operator can be seen as an extension of the CNOT port. It is considered universal and it is possible to build based on any other quantum operator. The chapter shows that if not handled efficiently, it would require a lot of memory space, which makes the design emulate a quantum processor quite expensive to produce.
This chapter discusses the execution of a quantum operation that manipulates a single quantum bit, showing the time diagram during its simulation. It illustrates the execution of an operation on non-entangled quantum bits, showing the time diagram during its simulation. When performing a quantum operation on two or more q-bits where at least one of them is entangled with another and/or when the quantum operation causes entanglement, the construction of the quantum operator by means of a tensor product is required. The coefficients of the basic quantum operators are read from memory MOP and stored in memory MSC when the operator is constructed for two q-bits. The construction of an operator for three or more q-bits requires the reading and writing of the coefficients in MSC, eventually occurring simultaneously, but always at different addresses.
This chapter represents our first step into quantum information theory (QIT). The key to operating such a transition is to become familiar with the concept of the quantum bit, or qubit , which is a probabilistic superposition of the classical 0 and 1 bits. In the quantum world, the classical 0 and 1 bits become the pure states |0〉 and |1〉, respectively. It is as if a coin can be classically in either heads or tails states, but is now allowed to exist in a superposition of both! Then I show that qubits can be physically transformed by the action of unitary matrices , which are also called operators . I show that such qubit transformations, resulting from any qubit manipulation, can be described by rotations on a 2D surface, which is referred to as the Bloch sphere . The Pauli matrices are shown to generate all such unitary transformations. These transformations are reversible , because they are characterized by unitary matrices ; this property always makes it possible to trace the input information carried by qubits. I will then describe different types of elementary quantum computations performed by elementary quantum gates, forming a veritable “zoo” of unitary operators, called I , X , Y , Z , H , CNOT, CCNOT, CROSSOVER or SWAP, controlled- U , and controlled-controlled- U . These gates can be used to form quantum circuits , involving any number of qubits, and of which several examples and tools for analysis are provided. Finally, the concept of tensor product , as progressively introduced through the above description, is eventually formalized.
In Chapter 15 , quantum emulations of some familiar invertible digital gates were discussed; specifically the not , cnot , Toffoli and Peres gates, each of which is significant in quantum computation, were considered. The emulations were defined on basis elements only – i.e., on https://www.w3.org/1998/Math/MathML" display="inline"> B ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_3.tif "/> or https://www.w3.org/1998/Math/MathML" display="inline"> B ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_4.tif "/> or https://www.w3.org/1998/Math/MathML" display="inline"> B ℂ 2 ⊗ ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_5.tif "/> etc. On these restricted inputs the quantum emulations were seen to produce outputs identical to their digital equivalents. Unlike the digital gates the quantum emulations may be extended beyond the basis sets to the entire tensor product spaces so defined – i.e., to https://www.w3.org/1998/Math/MathML" display="inline"> ℂ 2 , ℂ 2 ⊗ ℂ 2 , ℂ 2 ⊗ ℂ 2 ⊗ ℂ 2 https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003264569/7139ce19-0d58-4a0c-820f-b76fb033e438/content/math16_6.tif "/> etc. Such generalised inputs to the quantum gates lead to more general quantum outputs to which the simple deterministic quantum measurement processes of Chapter 15 are not applicable. The quantum gates defined on the extended domains are shown to be unitary, and are denoted in the same way as their restrictions to the basis vectors. We first consider a sufficient condition for an operator to be unitary; it has applications in our discussions below and in Chapter 17 .
Quantum computing (Q-computing) is a revolutionary technology that applies quantum mechanics (QM) to implement calculations significantly quicker than traditional computers. Q-computing employs Q-bits (qubits) rather than classical bits, which can take on a superposition state, making it possible to represent both 0 and 1 at once. This distinctive property allows qubits to perform multiple calculations simultaneously, making Q computers exponentially more powerful as more qubits are added, compared to the linear scaling of classical computers. However, challenges such as qubit instability and error rates remain significant obstacles to fully realizing this potential. A vital aspect of this technology is the development of Q arithmetic circuits, which act as the foundational blocks for efficiently executing Q algorithms. Q half-adder and full-adder circuits are integral to performing fundamental arithmetic operations within Q computers. In contrast to classical adders, Q adders rely on Q gates such as Controlled NOT (CNOT) and Toffoli gates. These gates exploit Q phenomena like entanglement and superposition, allowing for more efficient computations. The Q half-adder consists of two qubits and adds two single bits, producing a carry and sum output. It utilizes the CNOT gate to compute the sum and the Toffoli gate (controlled-controlled-NOT [CCNOT]) to compute the carry. The Q full-adder builds on this, adding three qubits to produce a sum and a carry output, further increasing computational capacity. Through the entanglement of qubits, these circuits allow for more parallelism and speed compared to classical adders. Q carry-look-ahead adders (QCLAs) and Q ripple-carry adders (QRCAs) are used to perform more complex arithmetic operations based on these half-adder and full-adder structures. Q arithmetic circuits have broad applications, boosting secure key generation in cryptography, speeding up matrix operations for faster model training in machine learning, and enabling more precise, efficient image transformations in Q image processing, benefiting fields like medical imaging and computer vision.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
We have seen how information can be encoded onto a quantum system by selecting the state in which it is prepared. Retrieving the information is achieved by performing a measurement, and the optimal measurement in any given situation is usually a generalized measurement. In between preparation and measurement, the information resides in the quantum state of the system, which evolves in a manner determined by the Hamiltonian. The associated unitary transformation may usefully be viewed as quantum information processing; if we can engineer an appropriate Hamiltonian then we can use the quantum evolution to assist in performing computational tasks. Our objective in quantum information processing is to implement a desired unitary transformation. Typically this will mean coupling together a number, perhaps a large number, of qubits and thereby generating highly entangled states. It is fortunate, although by no means obvious, that we can realize any desired multiqubit unitary transformation as a product of a small selection of simple transformations and, moreover, that each of these need only act on a single qubit or on a pair of qubits. The situation is reminiscent of digital electronics, in which logic operations are decomposed into actions on a small number of bits. If we can realize and control a very large number of such operations in a single device then we have a computer. Similar control of a large number of qubits will constitute a quantum computer. It is the revolutionary potential of quantum computers, more than any other single factor, that has fuelled the recent explosion of interest in our subject. We shall examine the remarkable properties of quantum computers in the next chapter. In digital electronics, we represent bit values by voltages: the logical value 1 is a high voltage (typically +5 V) and 0 is the ground voltage (0 V). The voltage bits are coupled and manipulated by transistor-based devices, or gates. The simplest gates act on only one bit or combine two bits to generate a single new bit, the value of which is determined by the two input bits. For a single bit, with value 0 or 1, the only possible operations are the identity (which does not require a gate) and the bit flip.