gottesman <Ch18

Author: valbert4

codes/quantum/qubits/qubits_into_qubits.yml

@@ -171,6 +171,7 @@ features:
         Circuit and measurement designs have to take care of the few stabilizer generators with large weights in order to be fault tolerant, but measurement duration may not pose a threat to scalability \cite{arxiv:quant-ph/0607047}.
         While generic concatenated methods yield a computational threshold with overhead \(O(M\text{polylog}M)\), concatenations using quantum Hamming codes can additionally attain constant space overhead with quasi-polylogarithmic time overhead \cite{arxiv:2207.08826,arxiv:2402.09606}, and concatenations of the Steane code and certain QLDPC codes further improve this time overhead to polylogarithmic while keeping constant space overhead \cite{arxiv:2411.03683}.
         Subsequently, thresholds were determined for infinite families of lattice stabilizer codes, starting with the toric code \cite{arxiv:quant-ph/0110143}; such a threshold is colloquially called a \textit{topological threshold}.
+        When different classes of circuit locations have different error rates, the single-number threshold generalizes to a \textit{threshold surface} in the space of error-rate vectors. A one-level crossing where a particular logical location becomes more reliable is then not a full-fledged threshold for the entire circuit; deciding whether a protocol really improves may require following the coupled recurrence relations through additional concatenation levels \cite[Sec. 14.7.6]{preset:GottesmanBook}.
         Fault-tolerant computations with no notion of locality can be made local on a 2D or 3D geometry with minimal overhead \cite{arxiv:2402.13863}.
         \end{defterm}
     - 'There is an upper bound on the threshold under local update recovery that is derived via quantum optimal transport \cite{arxiv:quant-ph/0310136} (see also Ref. \cite{arxiv:2309.16241}).'

codes/quantum/qubits/small_distance/quantum_repetition.yml

@@ -56,6 +56,7 @@ features:
 
   fault_tolerance:
     - 'Fault-tolerant syndrome detection \cite{arxiv:quant-ph/0412168}.'
+    - 'An \(m\)-qubit GHZ state, i.e., qubit cat state, can serve as an ancilla for fault-tolerant measurement of a weight-\(m\) Pauli operator by coupling each ancilla qubit to one data qubit, applying Hadamards to the ancilla, and reading out the parity of the measurement outcomes \cite{preset:GottesmanBook}.'
     - 'Toffoli magic-state preparation protocol \cite{arxiv:2012.04108}.'
   code_capacity_threshold:
     - 'Independent \(X\) noise: \(50\%\) with RG decoder for quantum repetition code arranged on a 1D or 2D lattice \cite{arxiv:1708.09286}.'
@@ -69,9 +70,10 @@ features:
 realizations:
   - 'NMR: 2-qubit phase-flip code \cite{arxiv:quant-ph/9811068,arxiv:cs/0012017}; 3-qubit bit-flip code \cite{arxiv:quant-ph/0004030}; 3-qubit phase-flip code \cite{arxiv:quant-ph/9802018,arxiv:1108.4842}, with up to two rounds of error correction in liquid-state NMR \cite{arxiv:1109.4821}. Such codes were used to characterize noise \cite{arxiv:quant-ph/0610038}.'
   - 'Linear optics: 2-qubit phase-flip code \cite{arxiv:quant-ph/0502042}.'
-  - 'Trapped ions: 3-qubit bit-flip code by Wineland group \cite{doi:10.1038/nature03074}, and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations \cite{doi:10.1126/science.1203329}.
-  Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum \cite{arxiv:2305.03828}.
-  Multiple rounds of Steane error correction \cite{arxiv:2312.09745}.'
+  - |
+    Trapped ions: 3-qubit bit-flip code by Wineland group \cite{doi:10.1038/nature03074}, and 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations \cite{doi:10.1126/science.1203329}.
+    Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum \cite{arxiv:2305.03828}.
+    Multiple rounds of Steane error correction \cite{arxiv:2312.09745}.
   - |
     Superconducting circuits: 3-qubit phase-flip and bit-flip code by Schoelkopf group \cite{arxiv:1004.4324,arxiv:1109.4948}; 3-qubit bit-flip code \cite{arxiv:1411.5542}; 3-qubit phase-flip code up to 3 cycles of error correction \cite{arxiv:1508.01388}; IBM 15-qubit device \cite{arxiv:1709.00990}; IBM Rochester device using 43-qubit code \cite{arxiv:2004.11037}; Google system performing up to 8 error-correction cycles on 5 and 9 qubits \cite{arxiv:1411.7403}; Google Quantum AI Sycamore utilizing up to 11 physical qubits and running 50 correction rounds \cite{arxiv:2102.06132}; Google Quantum AI Sycamore utilizing up to 25 qubits for comparison of logical error scaling with a quantum code \cite{arxiv:2207.06431} (see also \cite{arxiv:2211.04728}). 
     Google Quantum AI follow-up experiment on codes up to (classical) distance 29, demonstrating exponential suppression to an error floor of \(10^{-10}\) \cite{arxiv:2408.13687}. 

codes/quantum/qubits/small_distance/small/15/stab_15_1_3.yml

@@ -59,7 +59,7 @@ features:
     - 'Transversal logical \(CS\) gates between code blocks, together with transversal \(T\) and CNOT, can be used to realize IQP-like sampling architectures on a hypercube connectivity graph \cite{arxiv:2404.19005}.'
 
   general_gates:
-    - 'Code is often used in magic-state distillation protocols because of its transversal \(T\) gate \cite{arxiv:quant-ph/0403025}.'
+    - 'Code is often used in magic-state distillation protocols because of its transversal \(T\) gate \cite{arxiv:quant-ph/0403025}. In the original 15-to-1 protocol, 15 noisy \(R_{\pi/8}\ket{+}\) magic states are used to implement the transversal \(T\) by compressed gate teleportation, the resulting encoded state is post-selected by 15-qubit error detection, and decoding yields an output magic state with error probability \(O(p^3)\) when the input states have independent error probability \(p\) \cite[Sec. 13.5.1, Protocol 13.2]{preset:GottesmanBook}.'
     - 'Non-transversal logical Hadamard \cite{arxiv:2510.08402}.'
 
   decoders:

codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml

@@ -65,16 +65,16 @@ features:
     - 'Transversal gates can be interpreted as monodromies under a particular notion of parallel transport \cite[Exam. 6.4.2]{arxiv:1309.7062}.'
 
   general_gates:
-    - 'Magic-state distillation protocol \cite{arxiv:quant-ph/0403025}.'
+    - 'Magic-state distillation protocol \cite{arxiv:quant-ph/0403025}. One protocol distills the state \(\ket{R}=\cos\beta\ket{0}+e^{\mathrm{i}\pi/4}\sin\beta\ket{1}\), with \(\cos(2\beta)=1/\sqrt{3}\), using the fact that a transversal Clifford gate \(R\) is a gadget for the code; the protocol projects five noisy \(\ket{R}\) states onto the code space and suppresses the output error to \(O(p^2)\) for independent input error probability \(p\) \cite[Sec. 13.5.3, Protocol 13.5]{preset:GottesmanBook}.'
     - 'Pieceable fault-tolerant CZ, CNOT, and \(CCZ\) gates \cite{arxiv:1603.03948}.'
   decoders:
+    - 'Ideal transversal computational-basis measurement distinguishes logical basis states by the parity of the outcome string, but this is not a fault-tolerant measurement gadget because a single faulty measurement bit can flip the decoded logical outcome \cite[Sec. 11.4]{preset:GottesmanBook}.'
     - 'Fault-tolerant syndrome extraction circuits \cite{arxiv:quant-ph/9605031,arxiv:quant-ph/9608028}.'
     - 'Syndrome extraction circuit optimized for a linear qubit architecture \cite{arxiv:quant-ph/0311116}.'
     - 'Combined dynamical decoupling and error correction protocol on individually-controlled qubits with always-on Ising couplings \cite{arxiv:1509.01239}.'
     - 'Syndrome extraction circuit using only CNOT-SWAP gates \cite{arxiv:2207.13356}.'
     - 'Symmetric decoder correcting all weight-one Pauli errors. The resulting logical error channel after coherent noise has been explicitly derived \cite{arxiv:2203.01706}.'
     - 'Inspired by the honeycomb Floquet code, various weight-two measurement schemes have been designed \cite{arxiv:2409.13681}.'
-    - 'Ideal transversal computational-basis measurement distinguishes logical basis states by the parity of the outcome string, but this is not a fault-tolerant measurement gadget because a single faulty measurement bit can flip the decoded logical outcome \cite{preset:GottesmanBook}.'
 
   fault_tolerance:
     - 'Pieceable fault-tolerant CZ, CNOT, and \(CCZ\) gates \cite{arxiv:1603.03948}.'

codes/quantum/qubits/stabilizer/qldpc/concatenated/concatenated_steane.yml

@@ -29,6 +29,7 @@ features:
       - '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:
       - 'Between \(1.78\%\) and \(11.5\%\) with faulty photon detectors when combined with the dual-rail code at the first concatenation step in a variant of the KLM protocol \cite{arxiv:quant-ph/0405112,arxiv:quant-ph/0502101}.'
+      - 'For the adversarial-stochastic exRec analysis of the concatenated 7-qubit protocol, a crude bound gives \(p_T > 3.6\times 10^{-6}\), while circuit optimization together with careful counting of malignant sets improves this to \(p_T \geq 2.7\times 10^{-5}\) \cite[Sec. 14.7.4]{preset:GottesmanBook}.'
       - 'When used as the underlying code of a Steane/Hamming concatenation in a unified logical-CNOT comparison under circuit-level depolarizing noise, the threshold is \(0.030\%\); at physical error rate \(0.1\%\), this underlying code cannot suppress the logical error rate to \(10^{-24}\), while at \(0.01\%\) it requires space overhead \(6.1\times 10^3\) \cite{arxiv:2402.09606}.'
       - 'The recursively concatenated Steane code has a \hyperref[topic:measurement-threshold]{measurement threshold} of one \cite{arxiv:2402.00145}.'
 

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -131,7 +131,9 @@ features:
     - 'Fault-tolerant CNOT gate using generalized lattice surgery \cite{arxiv:2505.01370}.'
   fault_tolerance:
     - 'Steane error correction \cite{arxiv:quant-ph/9611027}, where fault-tolerance is ensured by preparing ancillary encoded states and extracting syndromes via \(CNOT\) gates.'
-    - 'Transversal computational-basis measurement followed by classical decoding is a fault-tolerant gadget for logical measurement of all encoded qubits \cite{preset:GottesmanBook}.'
+    - 'Encoded \(\ket{0}\) and \(\ket{+}\) ancillas for Steane error correction can be prepared by hierarchical Steane-style verification; without code-specific optimizations, a two-level procedure uses at least \((t+1)^2\) noisy ancillas for a distance-\(2t+1\) CSS code \cite[Secs. 13.1.2-13.1.3]{preset:GottesmanBook}.'
+    - 'Steane''s method also yields non-destructive logical Pauli measurement for CSS codes by coupling the data block transversally to encoded \(\ket{0}\) or \(\ket{+}\) ancillas and classically decoding the ancilla measurement results \cite{preset:GottesmanBook}.'
+    - 'Transversal computational-basis measurement followed by classical decoding is a fault-tolerant gadget for logical measurement of all encoded qubits \cite[Sec. 11.4]{preset:GottesmanBook}.'
     - 'Fault-tolerant error correction and logical measurements using flag qubits for distance-three cyclic CSS codes \cite{arxiv:1803.09758}. Parallel syndrome extraction for distance-three codes can be done fault-tolerantly using one flag qubit \cite{arxiv:2208.00581}. \hyperref[topic:effective-distance]{Distance-preserving} flag fault-tolerant error correction can be done using lookup tables for small codes \cite{arxiv:2306.12862}.'
     - 'Homomorphic gadgets fault-tolerant measurement unify Steane and Shor error correction \cite{arxiv:2211.03625}.'
     - 'A fault-tolerant error-correction protocol using \(O(d\log d)\) syndrome measurements can be applied to any CSS code with distance \(d \geq \Omega(n^{\alpha})\) for any \(\alpha > 0\) \cite{arxiv:2002.05180}.'

codes/quantum/qubits/stabilizer/qubit_stabilizer.yml

@@ -207,6 +207,7 @@ features:
 
   decoders:
     - 'The size of the circuit extracting the syndrome depends on the weight of its corresponding stabilizer generator. Syndrome extraction circuits can be simulated efficiently using dedicated software (e.g., STIM \cite{arxiv:2103.02202}) and there are many general schemes for generating them \cite{arxiv:2408.01339} (see also \cite{arxiv:2402.04093}). Noise can be characterized without interrupting syndrome extraction \cite{arxiv:1710.03636}. Decoding of qubit stabilizer codes is an approximately optimal strategy for various quantum lights-out (QLO) games that can be played on the codes'' \hyperref[topic:encoder-respecting]{encoder-respecting form} \cite{arxiv:2411.14448}.'
+    - 'Instead of applying the Pauli recovery from a fault-tolerant syndrome-extraction gadget, one can usually track it classically as a \textit{Pauli frame}; Clifford operations update the frame by conjugation, while measurement-based non-Clifford gadgets must be interpreted relative to the current frame \cite{preset:GottesmanBook}.'
     - 'A greedy graph decoder for \hyperref[topic:encoder-respecting]{encoder-respecting forms} corrects all recoverable errors for sufficiently sparse graphs, including graphs with girth at least 13 and some lower-girth families with additional spacing constraints \cite{arxiv:2411.14448}.'
     - 'DiVincenzo-Aliferis syndrome extraction circuits \cite{arxiv:quant-ph/0607047}.'
     - 'Greedy syndrome measurement schedule \cite{arxiv:2409.14283}.'
@@ -238,6 +239,7 @@ features:
     - 'Shor error correction \cite{arxiv:quant-ph/9605011,arxiv:quant-ph/9605031} (see also Steane''s ancilla factory \cite{arxiv:quant-ph/9708021}), in which fault tolerance against syndrome extraction errors is ensured by simply repeating syndrome measurements. A modification uses adaptive measurements \cite{arxiv:2208.05601}.'
     - 'Generalization of Steane error correction for stabilizer codes \cite[Sec. 3.6]{preset:Yoder18}.'
     - 'Fault-tolerant error correction scheme by Knill (a.k.a. telecorrection \cite{arxiv:quant-ph/0601066}), which is based on teleportation \cite{arxiv:quant-ph/0410199,arxiv:quant-ph/0312190}. A variant of it has been termed the Fibonacci scheme \cite{arxiv:0809.5063}.'
+    - 'For a non-CSS stabilizer code, the logical Bell-state ancilla needed for Knill error correction can be prepared by treating two code blocks as one larger stabilizer state and verifying the combined stabilizer via the Shor-style preparation procedure \cite[Sec. 13.1.3]{preset:GottesmanBook}.'
     - 'Fault-tolerant error correction using flag qubits for codes satisfying certain conditions \cite{arxiv:1708.02246}.'
     - 'GHZ state distillation for Steane error correction \cite{arxiv:2109.06248}.'
     - 'Syndrome extraction using flag qubits and classical codes \cite{arxiv:2212.10738}.'