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valbert4 committed Jan 14, 2025
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1 change: 1 addition & 0 deletions codes/quantum/properties/block/block_quantum.yml
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Expand Up @@ -40,6 +40,7 @@ protection: |
Block codes protect from erasures or, more generally, errors acting on a few of the \(n\) subsystems. A block code with \textit{distance} \(d\) detects errors acting on up to \(d-1\) subsystems, and corrects erasure errors on up to \(d-1\) subsystems.
The subsystems that are erased are known to the receiver, and erasures of subsystems at unknown locations are called \textit{deletion errors} \cite{arxiv:2001.08405,arxiv:2004.00814,arxiv:2102.02494,arxiv:2102.03015}.
More general forms of noise are caused by \textit{insertion errors} \cite{arxiv:2001.08405,arxiv:2004.00814,arxiv:2102.02494,arxiv:2102.03015}, where subsystems are inserted into the block, and \textit{synchronization errors} (a.k.a. misalignment) \cite{arxiv:1206.0260}, where the code block is misplaced in a larger block by one or more locations.
There are relations between deletion and insertion errors \cite{arxiv:2105.07214,arxiv:2501.07027}.
The \textit{weight} of an operator on a tensor-product Hilbert space is the number of subsystems on which the operator acts non-trivially.
For example, an operator acting on two subsystem is called a weight-two operator or a two-body operator.
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1 change: 1 addition & 0 deletions codes/quantum/properties/quantum_random.yml
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Expand Up @@ -10,6 +10,7 @@ name: 'Random quantum code'

description: 'Quantum code whose construction is non-deterministic in some way, i.e., codes that utilize an elements of randomness somewhere in their construction. Members of this class range from fully non-deterministic codes (e.g., random-circuit codes), to codes whose multi-step construction is deterministic with the exception of a single step (e.g., expander lifter-product codes).'


protection: |
Certain random codes have nontrivial \hyperref[topic:codespace-complexity]{codespace complexity} \cite{arxiv:2310.04710}.
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4 changes: 2 additions & 2 deletions codes/quantum/qubits/dynamic/random/haar_random.yml
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Expand Up @@ -17,10 +17,10 @@ description: |
Intuitively, coupling with the environment can be decreased by projecting the system onto a random codespace. The more qubits that are randomly discarded, the more the codespace is decoupled from the environment. One may ask what is the least amount of qubits that can be discarded, i.e. the largest remaining codespace, that still achieves decoupling. It can be shown through the decoupling inequality \cite{arxiv:quant-ph/0512247} that the largest possible dimension of the random codespace that achieves arbitrarily large decoupling is exponential in the coherent information of the channel. Therefore, there exist codes that can transmit information with rate equal to the coherent information. Furthermore, these codes can be constructed with high probability by performing a Haar-random isometry embedding a \(k\)-dimensional logical subspace into an \(n\)-dimensional physical space, where \(k/n\) is equal to the coherent information. Such an isometry can be produced by QR decomposition of a Gaussian random matrix \cite{doi:10.1137/0717034}.
protection: 'Random code achieve the capacity of any noisy quantum channel.'
features:
rate: 'The rate of the code is equal to the coherent information of the channel (i.e. the quantum channel capacity).'
rate: 'Haar-random qubit codes attain the regularized coherent information of certain noise channels in the limit of large \(n\) \cite{arxiv:quant-ph/0701102}.'

threshold:
- 'Haar-random qubit codes have a \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.'

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2 changes: 1 addition & 1 deletion codes/quantum/qubits/qubit_concatenated.yml
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Expand Up @@ -27,7 +27,7 @@ features:
- 'Fault-tolerant message passing between devices \cite{arxiv:2408.05260}.'
threshold:
- |
The first methods to achieve a \hyperref[topic:computational-threshold]{concatenated threshold} against local stochastic noise use concatenated qubit stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.
The first methods to achieve a \hyperref[topic:computational-threshold]{concatenated threshold} against local stochastic noise use concatenated qubit stabilizer codes \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0703230,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.
relations:
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4 changes: 2 additions & 2 deletions codes/quantum/qubits/qubits_into_qubits.yml
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Expand Up @@ -152,8 +152,8 @@ features:
\begin{defterm}{Computational threshold}
\label{topic:computational-threshold}
A fault-tolerant computational threshold is the maximum noise rate in a particular single-parameter noise model below which any logical computation of size \(M\) can be executed on a physical-qubit architecture to arbitrary accuracy and with an overhead of \hyperref[topic:asymptotics]{order} \(O(M\text{polylog}M)\).
The first methods to achieve a computational threshold use recursively concatenated stabilizer code families \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; such a threshold is called a \textit{concatenated threshold}.
Initially proven under local stochastic noise, the concatenated threshold theorem also holds for various types of non-Markovian noise \cite{arxiv:quant-ph/0402104,arxiv:quant-ph/0504218,arxiv:quant-ph/0510231}.
The first methods to achieve a computational threshold use recursively concatenated stabilizer code families \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9705031,arxiv:quant-ph/9903099,arxiv:quant-ph/9906129,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0703230,arxiv:quant-ph/0604090}; such a threshold is called a \textit{concatenated threshold}.
Initially proven under local stochastic noise, the concatenated threshold theorem also holds for various types of non-Markovian noise \cite{arxiv:quant-ph/0402104,arxiv:quant-ph/0504218,arxiv:quant-ph/0703230,arxiv:quant-ph/0510231}.
The resulting concatenated code is highly \hyperref[topic:degeneracy]{degenerate}, with all but an exponentially small fraction of generators having small weights.
Circuit and measurement designs have to take care of the few stabilizer generators with large weights in order to be fault tolerant.
Concatenated methods require constant-space and polylogarithmic-time overhead, but concatenations using quantum Hamming codes improve this to quasi-polylogarithmic time \cite{arxiv:2207.08826,arxiv:2402.09606}, and concatenations of the Steane code and certain QLDPC codes further improve this to polylogarithmic time \cite{arxiv:2411.03683}.
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Expand Up @@ -21,10 +21,11 @@ features:
- 'There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code \cite{arxiv:2403.09978}.'
- 'Randomized compiling helps reduce logical error rate for some noise models \cite{arxiv:2303.06846}.'
fault_tolerance:
- 'Fault-tolerant computation can be done on nearest-neighbor arrays \cite{arxiv:quant-ph/0702201}.'
- 'There exist fault-tolerant syndrome extraction protocols for the concatenated Steane code \cite{arxiv:2403.09978}.'
- 'The combination of the concatenated Steane code and QLDPC codes with non-vanishing rate yield fault-tolerant quantum computation with constant space and polylogarithmic time overheads, even when classical computation time is taken into account \cite{arxiv:2411.03683}.'
code_capacity_threshold:
- 'This family is one of the first to admit a \hyperref[topic:computational-threshold]{concatenated threshold} \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.'
- 'This family is one of the first to admit a \hyperref[topic:computational-threshold]{concatenated threshold} \cite{arxiv:quant-ph/9702058,arxiv:quant-ph/9809054,arxiv:quant-ph/0207119,arxiv:quant-ph/0410047,arxiv:quant-ph/0504218,arxiv:quant-ph/0703230,arxiv:quant-ph/0604090}; see the book \cite{preset:GottesmanBook}.'
threshold:
- 'Numerical study of \hyperref[topic:computational-threshold]{concatenated thresholds} of logical CNOT gates for various codes against depolarizing noise \cite{arxiv:0711.1556}; see also \cite{arxiv:quant-ph/0406025}.'
- 'A \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.'
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