Linear algebra helps as the mathematical language for quantum computing. It provides the framework to operate quantum states and operations. The basic concepts of linear algebra are critical for acquiring the principles behind quantum computation.
Quantum States as Vectors
In classical computing, the basic unit of information is a bit, which can be either 0 or 1. In quantum computing, the basic unit is a qubit, which, different from a classical bit, can exist in a superposition of states. A qubit’s state can be represented as a linear combination of two orthonormal basis states, denoted as |0⟩ and |1⟩. These basis for a two-dimensional complex vector space is known as a Hilbert space. Any state |ψ⟩ of a single qubit can be written as:
|ψ⟩ = α|0⟩ + β|1⟩
α and β are complex numbers representing the probability amplitudes of the qubit in the |0⟩ and |1⟩ states, respectively. The squared magnitudes of these amplitudes, |α|² and |β|², correspond to the probabilities of measuring the qubit in the |0⟩ or |1⟩ state, respectively, and satisfy the normalization condition
|α|² + |β|² = 1
This means the state of a qubit is a unit vector in a two-dimensional complex vector space.
With multiple qubits, their combined state is represented in a larger Hilbert space, which is the tensor product of the individual qubit spaces. For n qubits, the state space is 2n-dimensional. For example, the state of two qubits can be written as a linear combination of the basis states |00⟩, |01⟩, |10⟩, and |11⟩
|ψ⟩ = α|00⟩ + α|01⟩+ α|10⟩+ α|11⟩
As the number of qubits increases, the dimension of the state space grows exponentially (2n), allowing quantum computers to represent and potentially process huge amounts of information. This exponential growth in the state, enabled by the linear algebra framework, is a reason why quantum computers are believed to have the outperform classical computers for certain problems.
Quantum Operations as Linear Conversions
Quantum operations, which operate the states of qubits, are defined by linear conversions on the Hilbert space. These conversions also be unitary, which means they reserve the standard (length) of the state vector and are reversible. Mathematically, a quantum operation is denoted by a unitary matrix U, such that
U†U = I,
where U† is the conjugate transpose of U and I is the identity matrix.
Quantum gates are the building blocks of quantum circuits and implement these unitary conversions. Examples of single-qubit gates include the Pauli-X gate (bit-flip), Pauli-Z gate (phase-flip), and the Hadamard gate (creates superposition). These gates can be denoted by 2 times unitary matrices act on the state vector of a single qubit. Example, the Hadamard gate H is denoted by the matrix:
Applying the Hadamard gate to a qubit in the |0⟩ state (10) results in a superposition:
H|0⟩ = 1/√2 (|0⟩ + |1⟩)
Multi-qubit gates, such as the Controlled-NOT (CNOT) gate, operate on the combined state space of multiple qubits and are represented by larger unitary matrices (e.g., 4 times for two qubits). The CNOT gate, for example, flips the state of the second qubit (target) if and only if the first qubit (control) is in the |1⟩ state.
An arrangement of quantum gates forms a quantum circuit, and the overall operation of the circuit is defined by the product of the unitary matrices equivalent to each gate, applied in the reverse order of their execution.
Linear Algebra Concepts in Quantum Computing
Some concepts from linear algebra are important for understanding quantum computing:
- Vector Spaces: Quantum states reside in complex vector spaces (Hilbert spaces), where qubits are vectors in a two-dimensional space, and n-qubit systems are vectors in a 2n-dimensional space.
- Basis Vectors: The basis states (|0⟩, |1⟩ for a single qubit, and their tensor products for multiple qubits) form orthonormal bases for these vector spaces.
- Linear Independence and Span: Any quantum state can be stated as a linear combination of the basis vectors. A set of states is linearly independent if no state in the set can be written as a linear combination of the others. The basis vectors distance the entire vector space.
- Inner Product: The inner product (or dot product) between two quantum states 〈φ|ψ⟩ is a complex number that provides information about their overlap. If the inner product is zero, the states are orthogonal. The squared magnitude of the inner product 〈φ|ψ⟩2gives the probability of measuring state |ψ⟩ to be in state |φ⟩.
- Outer Product: The outer product of two vectors |ψ⟩ and |φ⟩ results in a linear operator (matrix) 〈ψ|φ⟩ Projection operators, which project a state onto a particular basis state, can be expressed using outer products.
- Eigenvalues and Eigenvectors: Eigenvectors of a linear operator are vectors that, when the operator is applied to them, are only scaled by a complex number called the eigenvalue. Eigenvalues and eigenvectors are critical for understanding the measurement process in quantum mechanics and the behaviour of quantum algorithms.
- Unitary Matrices: Quantum operations are represented by unitary matrices, which preserve the norm of quantum states, ensuring that probabilities remain consistent through the computation. The reversibility of quantum computation is linked to the unitarity of the operations.
- Tensor Product: The tensor product is used to combine the state spaces and operators of multiple quantum systems. If qubit 1 is in state |ψ1⟩ and qubit 2 is in state |ψ2⟩, their joint state is |ψ1⟩⊗|ψ2⟩ Similarly, the combined operation of two independent gates U1 and U2 acting on the respective qubits is U1⊗U2.
- Hilbert Spaces: Quantum states live in Hilbert spaces, which are complex vector spaces equipped with an inner product that allows for the definition of concepts like distance and angle between states. The finite-dimensional Hilbert spaces are sufficient for initial quantum computing concepts.
The quantum algorithms, such as Shor’s algorithm for factorization and Grover’s search algorithm, rises from the skill to operate these high-dimensional quantum state vectors using unitary connections in a way that controls quantum singularities like superposition and entanglement. For example, Shor’s algorithm depends on the Quantum Fourier Transform (QFT), which is a unitary transformation similar to the classical Discrete Fourier Transform and is capably implementable on a quantum computer. Grover’s algorithm uses a sequence of unitary operations to amplify the probability amplitude of the chosen state.
Linear algebra offers the mathematical language to describe the quantum world and to design the algorithms that attach its unique properties for computation. The concepts of vectors, matrices, unitary connections, and tensor products are fundamental tools for understanding how quantum computers store and process information. As quantum computing advances, a solid foundation in linear algebra will remain essential for researchers and practitioners in the field.