They can be zero or one, you can create them and change them and read them whenever you want without an existential crisis. The definitions we make here will motivate a lot of the rest of what we do, and is in my opinion one of the major hurdles to becoming comfortable with quantum computing.Ī bittersweet fact of life is that bits are comforting. But before we go romping about with what qubits can do, we need to understand how we can extract information from a qubit. We call such a vector a qubit.Ī qubit can assume “binary” values much like a regular bit, because you could pick two distinguished unit vectors, like and, and call one “zero” and the other “one.” Obviously there are many more possible unit vectors, such as and. That is, our most atomic unit of data is a vector with the constraints that are complex numbers and. The generalization of a bit is simple: it’s a unit vector in. In case you missed last time, our goal is to avoid as much physics as possible and frame everything purely in terms of linear algebra. Rather than go through that whole train of thought again let’s just jump into the definitions for the quantum setting. Indeed, in our motivational post we saw a glimpse of a probabilistic model of computation, where instead of the inputs being bits they were probabilities in a probability distribution, and instead of the gates being simple boolean functions they were linear maps that preserved probability distributions (we called such a matrix “stochastic”). If we want to come up with a different model of computing, we could start regular circuits and generalize some or all of these pieces. The V’s are “OR” and the Λ’s are “AND.” Image source: Ryan O’Donnell 'Show that the eigenvalues of a unitary matrix have modulus 1'.(2001), 'Unitary matrix', Encyclopedia of Mathematics, Springer Science+Business Media B.V. 'Restrictions on realizable unitary operations imposed by symmetry and locality' (in en). Physical Review A (American Physical Society (APS)) 52 (5): 3457–3467. 'Elementary gates for quantum computation'.

Margolus, Norman Shor, Peter Sleator, Tycho Smolin, John A. ↑ Barenco, Adriano Bennett, Charles H.Quantum Computation and Quantum Information. Linear Algebra and Its Applications 547: 32–44.

↑ Führ, Hartmut Rzeszotnik, Ziemowit (2018).Linear and Multilinear Algebra 50 (4): 321–326. 'Additive decomposition of real matrices'. Hermitian matrix and Skew-Hermitian matrix.Many other factorizations of a unitary matrix in basic matrices are possible.

This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ.

Given two complex vectors x and y, multiplication by U preserves their inner product that is, ⟨ U x, U y⟩ = ⟨ x, y⟩.

For any unitary matrix U of finite size, the following hold: