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Bloch sphere

The Bloch sphere (named after Felix Bloch) is a visual representation of the state of a qubit. Multiple states can also be simultaneously displayed (see below). The components of the Bloch representation of the qubit state are found from the expectation values of the X , Y , and Z operators. A qubit described by a state vector has unit length and is found on the surface of the Bloch sphere. Qubits characterized by a density matrix will in general have length less than one, as determined by the purity of the state, and lie within the Bloch sphere.


Evolution of a qubit state vector after applying a series of gates.


Entanglement is a property of quantum systems comprised of more than one subsystem (i.e., qubits), where the quantum state of any one subsystem cannot be uniquely described independently of the remaining subsystems. Mathematically an entangled state is one that can not be written as as product of subsystem states. Subsystems of entangled states are mixed states requiring a density matrix representation. For a bipartite quantum system, the entanglement is equal to the entropy of the subsystems.

global phase

A phase applied to a state vector as a whole, e^{i\gamma}|\psi\rangle . States related by a global phase are equivalent in quantum mechanics; global phases can be ignored. This is a consequence of the fact that only energy differences, as opposed to absolute values, matter in determining the dynamics of physical systems.

Grover’s algorithm

Grover’s algorithm solves the inverse problem for an input state, given a black box function (oracle) and a specified output. The solution is found quadratically faster than the best possible classical algorithm. Grover’s algorithm is an example of amplitude amplification.

quantum circuit

A quantum circuit is a computational routine consisting of coherent quantum operations on quantum data, such as qubits, and concurrent real-time classical computation. It is an ordered sequence of quantum gates, measurements and resets, all of which may be conditioned on and use data from the real-time classical computation.

quantum computer

A quantum computer is a device capable of executing coherent controlled quantum dynamics.

quantum gate

A quantum gate is a reversible (unitary) operation applied to one or more qubits.


A qubit (pronounced “cue-bit” and short for quantum bit) is the basic unit of quantum information. A qubit consists of two-levels that can be expressed using the “computational basis” states |0\rangle , and |1\rangle . Unlike a classical bit, the state of a qubit can be a linear combination (superposition) of both computational states.

relative phase

A phase difference between components of a superposition state. By convention, the first term in a superposition is made to be real, and the remaining states have phase values relative to this, e.g., |\psi\rangle=|0\rangle + e^{i\theta}|1\rangle .

Shor’s algorithm

A quantum algorithm that can factor integers in polynomial time, which is exponentially faster than the best-known classical algorithm. The techniques in Shor’s algorithm can also be used to compute discrete logarithms.


For a quantum computer comprised of a small number of qubits \sim\mathcal{O}(10) , we can simulate its behavior on a classical computer. In general such a computation requires storing 2^n complex numbers, where n is the number of qubits. For circuits composed solely of Clifford gates, or circuits generating quantum states that are weakly entangled, special simulation techniques allow for simulating a greater number of qubits.

state vector

Any single realization of a quantum system can be described through a complex vector known as its state vector. In a gate-based quantum computer the state of n qubits has 2^{n} elements; the dimension of the state vector grows exponentially with n .


A superposition in quantum mechanics is a weighted sum, or linear combination, of two or more quantum states. A quantum computer with n qubits can exist in a superposition of all 2^n of its computational basis states |000...0\rangle , through |111...1\rangle . Exploiting this ability is fundamental to most quantum algorithms.

uncertainty principle

In quantum physics, we cannot simultaneously know two non-commuting variables (like the position and momentum of a particle). This implies that a quantum system in a perfectly definite state can be certain under one measurement and completely random under another.  Moreover, if a quantum system starts out in an arbitrary unknown state, no measurement can reveal complete information about that state; the more information the measurement reveals, the more the state is disturbed.  This is a underlying principle of quantum cryptography.

universal fault-tolerant quantum computer

A universal fault-tolerant quantum computer is the grand challenge of quantum computing. It is a device that can properly perform universal quantum operations using unreliable components. See also universal quantum computer.

universal quantum computer

A universal quantum computer is a machine that can simulate an arbitrary quantum state from an arbitrary initial quantum state. See also universal fault-tolerant quantum computer.