No Such Code: Certified Invariants for Quantum Codes, Circuits, and Measurement

Four questions sit at the base of fault-tolerant quantum computing: whether a code with given parameters can exist, whether two circuits denote the same operation, how much entropy a measurement irreversibly produces, and whether a compiled gate still means what it should. They are normally answered by four disjoint techniques. We show they are one technique. In each case the answer is an invariant—the object a governing transformation leaves fixed—and that invariant can be held as a certificate a proof kernel checks and an operator-native substrate computes. The instruments are the quantum weight enumerator, whose shadow variant yields what is, to our knowledge, the first machine-checked nonexistence certificate for a quantum error-correcting code (no pure [[4, 1, 3]] stabilizer code, reduced to an integer impossibility); the spider normal form, unique both as a linear map and as a content-addressed tree, at the minimal axiomatic footprint of a proof assistant; the measurement boundary defect, whose logical entropy equals a kernel-counted cohomology dimension; and the reduction receipt, round-tripping a certified gate word to bare metal on two independent substrates. Everything downstream of the explicitly labelled modeling step in each domain is exact integer or rational arithmetic. The common shape—find the invariant the transformation fixes, and read it off as a certificate—turns four hard quantum questions into arithmetic a machine can hold. Published by The Institute for Applied Ontological Mathematics.Four questions sit at the base of fault-tolerant quantum computing: whether a code with given parameters can exist, whether two circuits denote the same operation, how much entropy a measurement irreversibly produces, and whether a compiled gate still means what it should. They are normally answered by four disjoint techniques. We show they are one technique. In each case the answer is an invariant—the object a governing transformation leaves fixed—and that invariant can be held as a certificate a proof kernel checks and an operator-native substrate computes. The instruments are the quantum weight enumerator, whose shadow variant yields what is, to our knowledge, the first machine-checked nonexistence certificate for a quantum error-correcting code (no pure [[4, 1, 3]] stabilizer code, reduced to an integer impossibility); the spider normal form, unique both as a linear map and as a content-addressed tree, at the minimal axiomatic footprint of a proof assistant; the measurement boundary defect, whose logical entropy equals a kernel-counted cohomology dimension; and the reduction receipt, round-tripping a certified gate word to bare metal on two independent substrates. Everything downstream of the explicitly labelled modeling step in each domain is exact integer or rational arithmetic. The common shape—find the invariant the transformation fixes, and read it off as a certificate—turns four hard quantum questions into arithmetic a machine can hold. Published by The Institute for Applied Ontological Mathematics.

Oct 11, 20260 citations

Ultra-high-distance quantum memories from amplified qLDPC codes

Zijian Liang, Boren Gu, Yu-An Chen +2 more

A large code distance does not by itself guarantee a well-protected quantum memory, as faults during syndrome extraction can propagate into correlated data errors. Certifying circuit distance becomes demanding as circuits grow. In this work, we introduce distance amplifiers, a modular approach to increasing both code distance and certified circuit-level protection in quantum low-density parity-check codes. The construction tensors a base code with an amplifier, adapting their syndrome-extraction circuits to the amplified code. To certify the resulting memory, we develop a fault-response certificate that yields rigorous lower bounds on circuit distance. Suitable gate ordering and flag qubits control correlated errors, allowing us to establish explicit conditions under which the amplified circuit distance equals the product of the constituent circuit distances. We prove this multiplicative relation for three types of amplifiers: the flagged $[[4,2,2]]$ , hypergraph-product, and rotated surface codes. The certificate extends recursively, so the circuit distance multiplies exactly at every amplification step. For example, four successive applications of the flagged $[[4,2,2]]$ amplifier to a $[[18,4,4]]$ seed yield a $[[13320,64,64]]$ code with the circuit distance $d_{\mathrm{circ}} \geq 48$ and check weights $w \leq 14$ . We further show how individual logical qubits can be tracked explicitly through amplification. By enabling high-distance quantum memories to be built and certified from smaller building blocks, our framework offers a systematic route toward scalable fault-tolerant quantum computing.A large code distance does not by itself guarantee a well-protected quantum memory, as faults during syndrome extraction can propagate into correlated data errors. Certifying circuit distance becomes demanding as circuits grow. In this work, we introduce distance amplifiers, a modular approach to increasing both code distance and certified circuit-level protection in quantum low-density parity-check codes. The construction tensors a base code with an amplifier, adapting their syndrome-extraction circuits to the amplified code. To certify the resulting memory, we develop a fault-response certificate that yields rigorous lower bounds on circuit distance. Suitable gate ordering and flag qubits control correlated errors, allowing us to establish explicit conditions under which the amplified circuit distance equals the product of the constituent circuit distances. We prove this multiplicative relation for three types of amplifiers: the flagged $[[4,2,2]]$, hypergraph-product, and rotated surface codes. The certificate extends recursively, so the circuit distance multiplies exactly at every amplification step. For example, four successive applications of the flagged $[[4,2,2]]$ amplifier to a $[[18,4,4]]$ seed yield a $[[13320,64,64]]$ code with the circuit distance $d_{\mathrm{circ}} \geq 48$ and check weights $w \leq 14$. We further show how individual logical qubits can be tracked explicitly through amplification. By enabling high-distance quantum memories to be built and certified from smaller building blocks, our framework offers a systematic route toward scalable fault-tolerant quantum computing.

quant-pharXiv:2609.37231Sep 30, 20260 citations

Quantum codes from trace codes

Pavan Kumar, Noor Mohammad Khan

Journal of Applied Mathematics and ComputingOct 23, 20220 citations

Constructing quantum codes

Hao Chen

Frontiers of Computer Science in ChinaJun 1, 20080 citations

Fault tolerance of quantum circuits with tensor networks and symplectic geometry

Soham Ghosh, Holger Boche, Andrew Tanggara

We develop an operator-algebraic framework for analyzing fault tolerance in quantum circuits and establish necessary and sufficient conditions for exact recovery of the ideal computation under a prescribed noise model. For prescribed circuit families and noise models, we derive a semidefinite program (SDP) based test whose infeasibility certifies that there does not exist any recovery map restoring its intended operation. For non-adaptive stabilizer circuits, analyzing the stabilizer symmetries of its components with symplectic geometry, we obtain closed-form detection and logical-effect matrices that characterize circuit distance algebraically. Applied to a finite Hastings--Haah honeycomb Floquet-code circuit, these matrices certify the dynamical encoding of two logical qubits and distance four under Pauli noise including measurement errors. Using these matrices, we then derive MacWilliams identities for circuit weight enumerators and obtain linear-programming (LP) upper bounds on circuit distance. Output-code constraints give additional Singleton-like bounds. Under independent and identically distributed (i.i.d.) depolarizing noise, the weight enumerators determine decoder failure probabilities and yield upper bounds on circuit pseudothresholds. Finally, we derive LP distance bounds for families of flag syndrome-extraction circuits specified by their CNOT orderings, by incorporating the $t$ -flag criterion as additional linear constraints to the distance-bounding LP. An analytical CSS Hamming-code benchmark yields a tight distance-three bound attained by the Chao--Reichardt one-flag construction.We develop an operator-algebraic framework for analyzing fault tolerance in quantum circuits and establish necessary and sufficient conditions for exact recovery of the ideal computation under a prescribed noise model. For prescribed circuit families and noise models, we derive a semidefinite program (SDP) based test whose infeasibility certifies that there does not exist any recovery map restoring its intended operation. For non-adaptive stabilizer circuits, analyzing the stabilizer symmetries of its components with symplectic geometry, we obtain closed-form detection and logical-effect matrices that characterize circuit distance algebraically. Applied to a finite Hastings--Haah honeycomb Floquet-code circuit, these matrices certify the dynamical encoding of two logical qubits and distance four under Pauli noise including measurement errors. Using these matrices, we then derive MacWilliams identities for circuit weight enumerators and obtain linear-programming (LP) upper bounds on circuit distance. Output-code constraints give additional Singleton-like bounds. Under independent and identically distributed (i.i.d.) depolarizing noise, the weight enumerators determine decoder failure probabilities and yield upper bounds on circuit pseudothresholds. Finally, we derive LP distance bounds for families of flag syndrome-extraction circuits specified by their CNOT orderings, by incorporating the $t$-flag criterion as additional linear constraints to the distance-bounding LP. An analytical CSS Hamming-code benchmark yields a tight distance-three bound attained by the Chao--Reichardt one-flag construction.

quant-pharXiv:2610.03429Oct 5, 20260 citations

Reliable Quantum Certification of Bosonic Code Preparations

WU Ya

Bosonic fault tolerant quantum computing requires preparations of Bosonic code states like cat states and GKP states with high fidelity and reliable quantum certification of these states. Although many proposals on preparing these states have been developed, few investigation has been done on how to reliably certify these experimental states. In this paper, we develop approaches to certify whether a continuous-variable state falls inside a certain Bosonic qubit code space by detecting a witness using Gaussian measurements. Our results can be applied to certification of various cat codes including two-component cat states, four-component cat states, and squeezed two-component cat states as well as Gottesman-Kitaev-Preskill codes. Then we further apply our approach to certify resource states in continuous-variable universal fault tolerant measurement-based quantum computing and quantum outputs of instantaneous quantum polynomial circuits, which can be used to show quantum supremacy. The sample complexity of our approach is efficient in terms of number of modes, so significantly reducing overhead compared to quantum state tomography in certification of many-mode quantum states.Bosonic fault tolerant quantum computing requires preparations of Bosonic code states like cat states and GKP states with high fidelity and reliable quantum certification of these states. Although many proposals on preparing these states have been developed, few investigation has been done on how to reliably certify these experimental states. In this paper, we develop approaches to certify whether a continuous-variable state falls inside a certain Bosonic qubit code space by detecting a witness using Gaussian measurements. Our results can be applied to certification of various cat codes including two-component cat states, four-component cat states, and squeezed two-component cat states as well as Gottesman-Kitaev-Preskill codes. Then we further apply our approach to certify resource states in continuous-variable universal fault tolerant measurement-based quantum computing and quantum outputs of instantaneous quantum polynomial circuits, which can be used to show quantum supremacy. The sample complexity of our approach is efficient in terms of number of modes, so significantly reducing overhead compared to quantum state tomography in certification of many-mode quantum states.

arXiv:2211.16777Nov 30, 20220 citations

[[4,2,2]]

, hypergraph-product, and rotated surface codes. The certificate extends recursively, so the circuit distance multiplies exactly at every amplification step. For example, four successive applications of the flagged

[[4,2,2]]

amplifier to a

[[18,4,4]]

seed yield a

[[13320,64,64]]

code with the circuit distance

d_{\mathrm{circ}} \geq 48

and check weights

w \leq 14

. We further show how individual logical qubits can be tracked explicitly through amplification. By enabling high-distance quantum memories to be built and certified from smaller building blocks, our framework offers a systematic route toward scalable fault-tolerant quantum computing.

t

-flag criterion as additional linear constraints to the distance-bounding LP. An analytical CSS Hamming-code benchmark yields a tight distance-three bound attained by the Chao--Reichardt one-flag construction.