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Hamiltonian-based code

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 for finite system size [6]. For infinite system size, Haah proved that every bounded local Hamiltonian evolution is a limit of a sequence of locality-preserving automorphisms (a.k.a. quantum cellular automata, or QCAs) [7; Thm. A.17].

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 [8].

A state \(|\psi\rangle\) in an invertible phase [9] 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 [10,11], and their low-energy excitations are often characterized by invertible field theories [12].

Protection

Ground states of many Hamiltonians can be easily written as tensor-network states or, in 1D, matrix product states (MPS). A no-go theorem states that open-boundary MPS that form a degenerate ground-state space of a gapped local Hamiltonian yield codes with distance that is only constant in the number of qubits \(n\), so MPS excitation ansatze have to be used to achieve a distance scaling nontrivially with \(n\) [13] (see also Ref. [14]).

Notes

Reviews of various quantum phases of matter and many-body systems [15–22].Book on rigorous results on stability of non-topological phases [23].

Cousins

  • Sphere packing— The Cohn-Elkies linear programming bound is related to the conformal bootstrap, which is a way of utilizing symmetry to constrain correlation functions of conformal field theories [24].
  • Low-density parity-check (LDPC) code— There are relations between LDPC codes and statistical mechanical models of spin glasses [25–28].
  • 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 \(n\) [29].
  • Quantum-double code— Quantum double code Hamiltonians can be simulated, with the help of perturbation theory and the \([[4,1,1,2]]\) subsystem code, by two-dimensional two-body Hamiltonians with non-commuting terms [30].
  • Pair-cat code— Two-legged pair-cat codewords form ground-state subspace of a multimode Kerr Hamiltonian.
  • Two-component cat code— The two-legged cat code forms the ground-state subspace of a Kerr Hamiltonian [31].
  • 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 [32].
  • Majorana box qubit— A Majorana box qubit form a fixed-parity subspace of the ground-state subspace of one or more Kitaev Majorana chain Hamiltonians.
  • Tetron code— Embedding each physical qubit into two fermions via the tetron code is useful for exactly solving the Kitaev honeycomb model Hamiltonian [33] and other qubit Hamiltonians on certain graphs [34,35]. Majorana stabilizer groups can be converted into ordinary qubit stabilizer groups via the parton mapping, while their corresponding states are converted via the Gutzwiller projection [36].
  • Concatenated qubit code— Concatenated stabilizer code Hamiltonians have been investigated [37].
  • 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 [38].
  • 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 [39].
  • 3D surface code— Stability of the 3D surface code against Hamiltonian perturbations was determined using a tensor-network representation [40]. 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 [41–43].

Member of code lists

Primary Hierarchy

Quantum Domain
Quantum code
Operator-algebra QECC (OAQECC)
Quantum error-correcting code (QECC)
Parents
Quantum error-correcting code (QECC)
Hamiltonian-based code
Children
Matrix-model code
Matrix-model codewords for simple codes are eigenstates of a matrix-model Hamiltonian.
Topological codeAbelian topological CDSC Kitaev surface
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 [44,45].
Commuting-projector Hamiltonian codeStabilizer CSS QLDPC Generalized homological-product Lattice stabilizer Bosonic stabilizer Analog stabilizer Quantum lattice Fracton stabilizer BB Homological Fermion-into-qubit Color Twist-defect color CDSC Twist-defect surface Kitaev surface Quantum QR code Quantum RM code QLTC
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 [44,46–49]. This notion can be extended to semi-hyperbolic manifolds [50] and non-geometrically local QLDPC codes exhibiting check soundness [51] (see also [52]). Hamiltonians admitting a Peierls condition are stable to off-diagonal perturbations [53].
Constant-excitation (CE) code
Constant-excitation codes are associated with a Hamiltonian governing the total excitations of the system.
Frustration-free Hamiltonian codeStabilizer CSS QLDPC Generalized homological-product Lattice stabilizer Bosonic stabilizer Analog stabilizer Quantum lattice Fracton stabilizer BB Homological Fermion-into-qubit Color Twist-defect color CDSC Twist-defect surface Kitaev surface Quantum QR code Quantum RM code QLTC
Symmetry-protected self-correcting quantum codeSelf-correcting quantum
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 [54].

References

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[2]
M. B. Hastings and X.-G. Wen, “Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance”, Physical Review B 72, (2005) arXiv:cond-mat/0503554 DOI
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T. J. Osborne, “Simulating adiabatic evolution of gapped spin systems”, Physical Review A 75, (2007) arXiv:quant-ph/0601019 DOI
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B. Nachtergaele, “From Lieb-Robinson Bounds to Automorphic Equivalence”, (2022) arXiv:2205.10460
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X. Chen, Z.-C. Gu, and X.-G. Wen, “Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order”, Physical Review B 82, (2010) arXiv:1004.3835 DOI
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S. Lloyd, “Universal Quantum Simulators”, Science 273, 1073 (1996) DOI
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J. Haah, “Invertible Subalgebras”, Communications in Mathematical Physics 403, 661 (2023) arXiv:2211.02086 DOI
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T. Schuster, J. Haferkamp, and H.-Y. Huang, “Random unitaries in extremely low depth”, (2025) arXiv:2407.07754
[9]
X.-G. Wen, “Colloquium : Zoo of quantum-topological phases of matter”, Reviews of Modern Physics 89, (2017) arXiv:1610.03911 DOI
[10]
Alexei Kitaev, Toward Topological Classification of Phases with Short-range Entanglement, talk at KITP (2011)
[11]
Alexei Kitaev, On the Classification of Short-Range Entangled States, talk at Simons Center (2013)
[12]
D. S. Freed, “Short-range entanglement and invertible field theories”, (2014) arXiv:1406.7278
[13]
M. Gschwendtner, R. König, B. Şahinoğlu, and E. Tang, “Quantum error-detection at low energies”, Journal of High Energy Physics 2019, (2019) arXiv:1902.02115 DOI
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B. Zeng, X. Chen, D.-L. Zhou, and X.-G. Wen, “Quantum Information Meets Quantum Matter -- From Quantum Entanglement to Topological Phase in Many-Body Systems”, (2018) arXiv:1508.02595
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S. Sachdev, “The quantum phases of matter”, (2012) arXiv:1203.4565
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E. H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics (Cambridge University Press, 2009) DOI
[24]
T. Hartman, D. Mazáč, and L. Rastelli, “Sphere packing and quantum gravity”, Journal of High Energy Physics 2019, (2019) arXiv:1905.01319 DOI
[25]
Franz, M. Leone, A. Montanari, and F. Ricci-Tersenghi, “Dynamic phase transition for decoding algorithms”, Physical Review E 66, (2002) arXiv:cond-mat/0205051 DOI
[26]
M. Mézard and A. Montanari, “Information, Physics, and Computation”, (2009) DOI
[27]
H. Nishimori, “Statistical Physics of Spin Glasses and Information Processing”, (2001) DOI
[28]
I. F. Blake, Essays on Coding Theory (Cambridge University Press, 2024) DOI
[29]
H. Apel and N. Baspin, “Simulating LDPC code Hamiltonians on 2D lattices”, (2023) arXiv:2308.13277
[30]
C. G. Brell, S. T. Flammia, S. D. Bartlett, and A. C. Doherty, “Toric codes and quantum doubles from two-body Hamiltonians”, New Journal of Physics 13, 053039 (2011) arXiv:1011.1942 DOI
[31]
S. Puri, S. Boutin, and A. Blais, “Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving”, npj Quantum Information 3, (2017) arXiv:1605.09408 DOI
[32]
T. Kohler and T. Cubitt, “Toy models of holographic duality between local Hamiltonians”, Journal of High Energy Physics 2019, (2019) arXiv:1810.08992 DOI
[33]
A. Kitaev, “Anyons in an exactly solved model and beyond”, Annals of Physics 321, 2 (2006) arXiv:cond-mat/0506438 DOI
[34]
A. Chapman and S. T. Flammia, “Characterization of solvable spin models via graph invariants”, Quantum 4, 278 (2020) arXiv:2003.05465 DOI
[35]
S. J. Elman, A. Chapman, and S. T. Flammia, “Free Fermions Behind the Disguise”, Communications in Mathematical Physics 388, 969 (2021) arXiv:2012.07857 DOI
[36]
R. A. Macedo, C. C. Bellinati, W. B. Fontana, E. C. Andrade, and R. G. Pereira, “Partons from stabilizer codes”, (2025) arXiv:2505.02683
[37]
D. Bacon, “Stability of quantum concatenated-code Hamiltonians”, Physical Review A 78, (2008) arXiv:0806.2160 DOI
[38]
M. Kargarian, H. Bombin, and M. A. Martin-Delgado, “Topological color codes and two-body quantum lattice Hamiltonians”, New Journal of Physics 12, 025018 (2010) arXiv:0906.4127 DOI
[39]
C. Fernández-González, N. Schuch, M. M. Wolf, J. I. Cirac, and D. Pérez-García, “Gapless Hamiltonians for the Toric Code Using the Projected Entangled Pair State Formalism”, Physical Review Letters 109, (2012) arXiv:1111.5817 DOI
[40]
D. J. Williamson, C. Delcamp, F. Verstraete, and N. Schuch, “On the stability of topological order in tensor network states”, Physical Review B 104, (2021) arXiv:2012.15346 DOI
[41]
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[42]
J. Dorier, F. Becca, and F. Mila, “Quantum compass model on the square lattice”, Physical Review B 72, (2005) arXiv:cond-mat/0501708 DOI
[43]
Z. Nussinov and J. van den Brink, “Compass and Kitaev models -- Theory and Physical Motivations”, (2013) arXiv:1303.5922
[44]
S. Michalakis and J. P. Zwolak, “Stability of Frustration-Free Hamiltonians”, Communications in Mathematical Physics 322, 277 (2013) arXiv:1109.1588 DOI
[45]
S. Bachmann, W. De Roeck, B. Donvil, and M. Fraas, “Stability of invertible, frustration-free ground states against large perturbations”, Quantum 6, 793 (2022) arXiv:2110.11194 DOI
[46]
S. Bravyi and M. B. Hastings, “A Short Proof of Stability of Topological Order under Local Perturbations”, Communications in Mathematical Physics 307, 609 (2011) arXiv:1001.4363 DOI
[47]
S. Bravyi, M. B. Hastings, and S. Michalakis, “Topological quantum order: Stability under local perturbations”, Journal of Mathematical Physics 51, (2010) arXiv:1001.0344 DOI
[48]
B. Nachtergaele, R. Sims, and A. Young, “Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms”, Journal of Mathematical Physics 60, (2019) arXiv:1810.02428 DOI
[49]
B. Nachtergaele, R. Sims, and A. Young, “Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States”, Annales Henri Poincaré 23, 393 (2021) arXiv:2010.15337 DOI
[50]
A. Lavasani, M. J. Gullans, V. V. Albert, and M. Barkeshli, “On stability of k-local quantum phases of matter”, (2024) arXiv:2405.19412
[51]
C. Yin and A. Lucas, “Low-density parity-check codes as stable phases of quantum matter”, (2024) arXiv:2411.01002
[52]
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[53]
M. Raginsky, “Almost any quantum spin system with short-range interactions can support toric codes”, Physics Letters A 294, 153 (2002) arXiv:quant-ph/0111035 DOI
[54]
R. Movassagh and Y. Ouyang, “Constructing quantum codes from any classical code and their embedding in ground space of local Hamiltonians”, Quantum 8, 1541 (2024) arXiv:2012.01453 DOI
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Zoo Code ID: hamiltonian

Cite as:
“Hamiltonian-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamiltonian
BibTeX:
@incollection{eczoo_hamiltonian, title={Hamiltonian-based code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/hamiltonian} }
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“Hamiltonian-based code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/hamiltonian

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/hamiltonian/hamiltonian.yml.