Math Foundations
Eigenstate
A state that a measurement or operation leaves unchanged except for a number, such as an energy or a phase.
Definition
An eigenstate is a special state that stays in the same direction when a particular operation is applied, apart from being scaled by a number. For a measurement, an eigenstate is a state that always gives the same answer. For the energy operator, an eigenstate is a state with a definite energy. Quantum algorithms such as phase estimation work by finding the number attached to an eigenstate.
In depth
Every observable has a set of eigenstates, and measuring the observable always returns one of its eigenvalues. If the system is already in an eigenstate, the result is certain. If it is in a superposition of eigenstates, the result is random, with probabilities set by the squared amplitudes, and the system then collapses into the eigenstate that was found. This is why the choice of measurement basis matters so much in quantum circuits.
The eigenstates of the Hamiltonian are the stationary states of a system, called energy eigenstates. A molecule's ground state is its lowest-energy eigenstate, and finding it is the central task of quantum chemistry. An energy eigenstate changes with time only by a phase, which is why it stays put while superpositions of energy eigenstates evolve and interfere.
Eigenstates are the target of several major algorithms. Quantum phase estimation reads out the eigenvalue of a unitary when it is given one of its eigenstates, the variational quantum eigensolver searches for the lowest-energy eigenstate, and Shor's algorithm uses phase estimation on an operator whose eigenvalues encode the period of a function. Preparing a good approximation to the right eigenstate is often the hardest step.
Technical Definition
If A is an operator and A times the state psi equals a times psi for a number a, then psi is an eigenstate of A with eigenvalue a. For a Hermitian operator, eigenvalues are real and eigenstates for different eigenvalues are orthogonal. For a unitary operator, eigenvalues have the form e to the i theta, a pure phase.
Visual Explanation: An Analogy
Think of a guitar string. Most ways of plucking it produce a messy mix of tones, but a few special vibration patterns, the harmonics, keep their shape and only oscillate at one pure frequency. Those patterns are eigenstates, and their frequencies are eigenvalues.
Key facts
| Item | Detail |
|---|---|
| Defining equation | A psi equals a psi, where a is the eigenvalue |
| Hermitian operators | Real eigenvalues and orthogonal eigenstates for different eigenvalues |
| Unitary operators | Eigenvalues are phases of magnitude 1 |
| Computational basis | The states 0 and 1 are eigenstates of the Pauli Z operator with eigenvalues plus 1 and minus 1 |
| Ground state | The lowest-energy eigenstate of the Hamiltonian |
Where the idea comes from
The eigenvalue idea comes from linear algebra and the study of vibrating strings. Heisenberg's matrix mechanics and Schrödinger's wave mechanics in 1925 and 1926 both turned on finding the eigenvalues and eigenstates of the energy operator, which explained the discrete spectra of atoms.
Real-World Use Cases
- Choosing measurement bases, since measuring in the eigenbasis of an operator gives definite outcomes.
- Finding ground states of molecules and materials with VQE and phase estimation.
- Encoding answers in phases, as in Shor's algorithm and amplitude estimation.
- Understanding stationary states and energy levels in quantum simulation.
How it differs from related ideas
- An eigenstate gives a definite result for one particular observable, and may be a superposition with respect to others.
- An eigenvalue is a number, and an eigenstate is the state that carries it.
- A computational basis state is an eigenstate of Z, but a superposition with respect to X.
Common Misconceptions
- Being an eigenstate is relative to an operator. A state can be an eigenstate of one observable and not another.
- An eigenstate is not necessarily a stable or low-energy state. It is just a state with a definite value of some quantity.
- Eigenvalues are not probabilities. They are the possible measurement results.
Frequently asked questions
What is an eigenstate in simple words?
A state that a given operation leaves pointing in the same direction, only multiplied by a number. For a measurement, it is a state that always gives the same result.
Why are eigenstates important for quantum algorithms?
Because algorithms such as phase estimation read out the eigenvalue attached to an eigenstate, and many problems, such as finding a molecule's energy, are eigenvalue problems.
Is the state 0 an eigenstate?
Yes, of the Pauli Z operator, with eigenvalue plus 1. It is not an eigenstate of the Pauli X operator.
Related Terms
Keep exploring
- Algorithm: Quantum Phase Estimation (QPE)
- Algorithm: Variational Quantum Eigensolver (VQE)
- Hardware: Google Sycamore