## Description

Code whose codespace corresponds to a set of energy eigenstates of a quantum-mechanical Hamiltonian i.e., a Hermitian operator whose expectation value measures the energy of its underlying physical system. The codespace is typically a set of low-energy eigenstates or ground states, but can include subspaces of arbitrarily high energy. Hamiltonians whose eigenstates are the canonical basis elements are called classical; otherwise, a Hamiltonian is called quantum.

A Hamiltonian whose ground states minimize the energy of each term is called a frustration-free Hamiltonian. A Hamiltonian whose terms commute and can be written as orthogonal projectors (i.e., with eigenvalues zero or one) is called a commuting projector Hamiltonian.

For block quantum codes, the code Hamiltonian is typically written as a sum of terms, with each term acting on at most some number \(K\) of subsystems (i.e., being of weight at most \(K\)). When \(K\) is independent of the total number of subsystems (e.g., QLDPC codes), the Hamiltonian is called a \(K\)-body or \(K\)-local Hamiltonian. Otherwise, the Hamiltonian is called non-local. When the physical space is endowed with a geometry, the Hamiltonian is typically geometrically local, consisting of operators acting on subsystems that occupy a region whose size is independent of the number of subsystems.

The notion of locality can be softened to include Hamiltonians whose terms each act non-trivially on all sites, but whose support on regions farther and farther from some designated central site decaying super-polynomially with the radius from the center. Such Hamiltonians are called quasi-local Hamiltonians (a.k.a. almost local or approximately local).

### Quantum phases of matter

Ground states of infinite families geometrically local block-code Hamiltonians on tesselations of Euclidean geometries are said to be in a particular phase of quantum matter, i.e., a region in some parameter space of Hamiltonians "in which [ground] states possess properties that can be distinguished from those in other phases" [1]. For example, topological code Hamiltonians realize a particular phase called a topological phase.

Hamiltonians realizing different phases cannot be adiabatically deformed into one another without a closing of the energy gap between the ground and excited states. Such adiabatic deformations naively would be generated by non-local Hamiltonians. However, Hastings and Wen [2] (see also [3,4]) showed that adiabatic evolution can in fact be generated by a quasi-local operator; such evolution is often called quasi-adiabatic evolution, quasi-adiabatic continuation, or spectral flow.

Two states in the same phases can be deformed into one another by evolving, via quasi-adiabatic evolution, for a time independent of the system size \(n\) [5; Appendix]. The unitary operation generated by a quasi-local Hamiltonian can be simulated by a quantum circuit, with the time of evolution determining the depth of the circuit. Approximating such constant-time evolution with constant error using a quantum circuit can be done in constant depth [6].

States in certain phases (e.g., topological phases) remain in said phases even after evolving for longer times (e.g., times that are logarithmic in \(n\)). This means that circuits of non-constant depth may leave a state in the same phase, making the phase classification problem quantum computationally hard [7].

A state \(|\psi\rangle\) in an invertible phase [8] can be deformed to a product state if it is first tensored with other states in related invertible phases that are realized on copies of the underlying lattice of \(|\psi\rangle\). Invertible phases are sometimes referred to as short-range entangled phases [9,10], and their low-energy excitations are often characterized by invertible field theories [11].

## Protection

## Notes

## Parent

## Children

- Matrix-model code — Matrix-model codewords for simple codes are eigenstates of a matrix-model Hamiltonian.
- Topological code — Codespace of a topological code is typically the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase. Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes). Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the TQO conditions), meaning that a notion of a phase can be defined [23,24].
- Commuting-projector Hamiltonian code — Geometrically local commuting-projector code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the TQO conditions, meaning that a notion of a phase can be defined [23,25–28]. This notion can be extended to semi-hyperbolic manifolds [29].
- Constant-excitation (CE) code — Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.
- Frustration-free Hamiltonian code
- Symmetry-protected self-correcting quantum code
- Conformal-field theory (CFT) code — CFT codewords lie in the low-energy subspace of a conformal field theory (CFT), e.g., the quantum Ising model at its critical point.
- SYK code — The SYK code Hamiltonian is constructed out of non-commuting few-site terms, and every fermion participates in many interactions.
- Eigenstate thermalization hypothesis (ETH) code — ETH codewords are eigenstates of a local Hamiltonian whose eigenstates satisfy ETH, and many example codes are eigenstates of frsutration-free Hamiltonians.
- Movassagh-Ouyang Hamiltonian code — Movassagh-Ouyang codes reside in the ground space of a Hamiltonian. Justesen codes can be used to build a family of \(n\)-qudit Movassagh-Ouyang Hamiltonian spin codes encoding one logical qubit with linear distance. These codes form the ground-state subspace of a frustration-free geometrically local Hamiltonian [30].

## Cousins

- Low-density parity-check (LDPC) code — There are relations between LDPC codes and statistical mechanical models of spin glasses [31–33].
- Quantum-double code — Quantum double code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional two-body Hamiltonians with non-commuting terms [34].
- Two-component cat code — The two-legged cat code forms the ground-state subspace of a Kerr Hamiltonian [35].
- Pair-cat code — Two-legged pair-cat codewords form ground-state subspace of a multimode Kerr Hamiltonian.
- Error-corrected sensing code — Metrologically optimal codes admit a \(U(1)\) set of gates generated by a signal Hamiltonian \(H\), meaning that there exists a basis of codewords that are eigenstates of the \(H\).
- Holographic tensor-network code — Local Hamiltonians lying at the CFT boundary can be mapped into the AdS bulk using tools from Hamiltonian simulation theory [36].
- Abelian topological code — Subsystem stabilizer code Hamiltonians described by an Abelian anyon theory do not always realize the corresponding anyonic topological order in their ground-state subspace and may exhibit a rich phase diagram. For example, the Kitaev honeycomb Hamiltonian admits the anyon theories of the 16-fold way, i.e., all minimal modular extensions of the \(\mathbb{Z}_2^{(1)}\) Abelian non-chiral non-modular anyon theory [37][38; Footnote 25].
- Quantum LDPC (QLDPC) code — QLDPC code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional Hamiltonians with non-commuting terms whose interactions scale with \(n\) [39].
- Majorana box qubit — The tetron code forms the ground-state subspace of two Kitaev Majorana chain Hamiltonians.
- Concatenated qubit code — Concatenated stabilizer code Hamiltonians have been investigated [40].
- 2D color code — 2D color code Hamiltonians can be simulated, with the help of perturbation theory, by two-dimensional weight-two (two-body) Hamiltonians with non-commuting terms [41].
- Kitaev surface code — While codewords of the surface code form ground states of the code's stabilizer Hamiltonian, they can also be ground states of other gapless Hamiltonians [42].
- 3D surface code — Stability of the 3D surface code against Hamiltonian perturbations was determined using a tensor-network representation [43]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.
- Bacon-Shor code — The 2D Bacon-Shor gauge-group Hamiltonian is the compass model [44–46]. Bacon-Shor code Hamiltonians can be used to suppress errors in adiabatic quantum computation [47], while subspace-code Hamiltonians with weight-two (two-body) terms cannot [48].

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## Page edit log

- Victor V. Albert (2022-02-15) — most recent

## Cite as:

“Hamiltonian-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamiltonian