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Local stabilizer codes in 3D without string logical operators

Jeongwan Haah

Published Jan 10, 2011• 1 citation
OpenAlex

Abstract

We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string operators. Previously known local stabilizer codes in 3D all have string-like operators, which make the codes non-self-correcting. We introduce a notion of logical string segments to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code has surface-like operators whose partial implementation has unsatisfied stabilizers along its boundary.

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We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string operators. Previously known local stabilizer codes in 3D all have string-like operators, which make the codes non-self-correcting. We introduce a notion of logical string segments to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code has surface-like operators whose partial implementation has unsatisfied stabilizers along its boundary.
We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have stringlike logical operators, which make the codes non-self-correcting. We introduce a notion of ``logical string segments'' to avoid difficulties in defining one-dimensional objects in discrete lattices. We prove that every stringlike logical operator of our code can be deformed to a disjoint union of short segments, each of which is in the stabilizer group. The code has surfacelike logical operators whose partial implementation has unsatisfied stabilizers along its boundary.
Quantum error-correcting codes (QECs) are essential components of quantum computation and have deep connections to quantum phases of matter. A key obstruction to passive self-correcting QECs is the presence of string logical operators, which can generate logical errors through constant-energy-barrier processes. Haah's Codes (fracton codes) showed that three-dimensional stabilizer codes can forbid such string logical operators, but their translation-invariant structure supports self-similar fractal logical operators with a logarithmic energy barrier. We introduce the qutrit random cubic codes, a family of local qutrit Calderbank-Shor-Steane stabilizer Hamiltonians with similar cube-check structure as Haah's Code 1 but built from spatially varying stabilizers. We prove that these models retain the no-string property and numerically observe that they have properties distinct from translation-invariant fracton codes: the smallest ground-state degeneracy exponent is k=2k=2k=2 for odd LLL and k=4k=4k=4 for even LLL; noncontractible plane-logical operators span the entire logical space; and charge-push diagnostics show that the self-similar fractal operators are absent. These results demonstrate that constrained randomness can fundamentally change the nature of stabilizer codes and improve their self-correction properties. They further point to broader families of quantum error-correcting codes and quantum phases beyond canonical topological and fracton orders.
We study properties of stabilizer codes that permit a local description on a regular D -dimensional lattice. Specifically, we assume that the stabilizer group of a code (the gauge group for subsystem codes) can be generated by local Pauli operators such that the support of any generator is bounded by a hypercube of size O (1). Our first result concerns the optimal scaling of the distance d with the linear size of the lattice L . We prove an upper bound d = O ( L D -1 ) which is tight for D =1, 2. This bound applies to both subspace and subsystem stabilizer codes. Secondly, we analyze the suitability of stabilizer codes for building a self-correcting quantum memory. Any stabilizer code with geometrically local generators can be naturally transformed to a local Hamiltonian penalizing states that violate the stabilizer condition. A degenerate ground state of this Hamiltonian corresponds to the logical subspace of the code. We prove that for D =1, 2, different logical states can be mapped into each other by a sequence of single-qubit Pauli errors such that the energy of all intermediate states is upper bounded by a constant independent of the lattice size L . The same result holds if there are unused logical qubits that are treated as 'gauge qubits'. It demonstrates that a self-correcting quantum memory cannot be built using stabilizer codes in dimensions D =1, 2. This result is in sharp contrast with the existence of a classical self-correcting memory in the form of a two-dimensional (2D) ferromagnet. Our results leave open the possibility for a self-correcting quantum memory based on 2D subsystem codes or on 3D subspace or subsystem codes.
Recently Haah introduced a new quantum error correcting code embedded on a cubic lattice. One of the defining properties of this code is the absence of string logical operator. We present new codes with similar properties by relaxing the condition on the local particle dimension. The resulting code is well-defined when the local Hilbert space dimension is prime. These codes can be divided into two different classes: the local stabilizer generators are either symmetric or antisymmetric with respect to the inversion operation. These is a nontrivial correspondence between these two classes. For any symmetric code without string logical operator, there exists a complementary antisymmetric code with the same property and vice versa. We derive a sufficient condition for the absence of string logical operator in terms of the algebraic constraints on the defining parameters of the code. Minimal number of local particle dimension which satisfies the condition is 5. These codes have logarithmic energy barrier for any logical error.
We construct self-correcting 3D surface codes over Zq\mathbb Z_qZq​ with local dimension qqq growing with system size. The model preserves the local geometry and stabilizer support of the ordinary 3D surface code and remains translation invariant in the bulk, while modifying only the stabilizer Pauli powers. We use the Lee metric to characterize error weight and consider local Metropolis dynamics with single-qudit X±1X^{\pm1}X±1 and Z±1Z^{\pm1}Z±1 updates, each changing the energy by at most a constant independent of qqq and system size NNN. We prove that, below the critical temperature, when the local dimension scales as q=Θ(N)q=Θ(N)q=Θ(N), the memory times in both error sectors grow at least as exp⁡[Ω(N1/3)]\exp\left[Ω(N^{1/3})\right]exp[Ω(N1/3)]. Our numerical results further support these analytical results. Together, these results establish growing local Hilbert-space dimension as a new resource for designing self-correcting quantum memories while retaining simple local geometry and local thermal dynamics.
The following open problems, which concern a fundamental limit on coding properties of quantum codes with realistic physical constraints, are analyzed and partially answered here: (a) the upper bound on code distances of quantum error-correcting codes with geometrically local generators, (b) the feasibility of a self-correcting quantum memory. To investigate these problems, we study stabilizer codes supported by local interaction terms with translation and scale symmetries on a DDD-dimensional lattice. Our analysis uses the notion of topology emerging in geometric shapes of logical operators, which sheds a surprising new light on theory of quantum codes with physical constraints.