Previously
Threshold for markovian and nonlocal quantum gates were derived previously [aliferis2005]
Survey on Coherent Error
Simulation with realistic nosie is necessary for designing better error correction code. Noise can be broken down into two categories, incoherent noise coherent noise. Incoherent noise is stochastic while coherent noise is unitary. Amplitude damping, depolarizing, and phase damping are all examples of incoherent noise. Imperfect control in qubit gate implementation is an example of coherent noise. Quantum circuit with depolarizing noise can be efficiently simulated thanks to Gottesman-Knill Theorem. Therefore the nosie threshold is best understood in this regime. The study of the other noise faces the challange of exponentially increasing simulation cost. In this survey, we focus on coheret noise.
Summary
Three approaches exist for the study of coherent noise.
- Direct simulation with coherent noise.
- Verify deviation of coherent noise from depolarizing noise is small on logical qubit level and then simulate with Clifford-based methods.
- Analytic study of coherent noise.
Many consider quantum memory and not quantum computing. Focus was on simple code, i.e repetition code, or surface code. The former because it was small therefore managable in simulation. The later because it was easy to realize experimentally and have high noise threshold. See citation (9-11) in [bravyi2018].
Direct Simulation Approach
Tensor Network Approach
PEPS encoded physical qubit state was used to do non-Clifford simulation. However, the size of result is still limited to 153 data qubits and perfect measurement was assumed. In case of concatenated code, this is far from sufficient.[Darmawan2017]
Mapping to Statistical Model
In 2002, mapping to Ising Model (2+1D $\mathbb{Z}_2$ lattice gauge model? ) from Toric Code was proposed. [Dennis2001] Assuming incoherent and uncorrelated error with measurement error, an error threshold was derived by judging the phase transition in the Ising Model.
Almost concurrently, a work appeared on flawless measurement connecting to random-bond Ising Model was studied[Wang2002]
In 2009, work was done on topological color code [Katzgraber2009].
In 2018, partially correlated noise was considered [Chubb2018]. Why doesn't any one mention the code size?
By"map(ping) the complete evolution after one quantum error correction cycle onto the problem of computing correlation functions of a two-dimensional Ising model with boundary fields", [jouzdani2014] tries to find the noise threshold. However, they limit the error and gate to nearest neighbor.
Mapping to Fermionic Linear Operator
Convertting repetition code with coherent noise into matchgate circuit enables efficient simulation. [Suzuki2017] [bravyi2018] used FLO for simulation following suzuki. [venn2020] is the follow up of Bravyi's work. [venn2023] follow up to the [venn2020] paper, estimates error threshold with the help of mapping error correction code to majorana fermion. Working as quantum memory with readout error, appears similar to venn's work [marton2023]. Analyzes a slightly different kind of coherent error with FLO approach [pataki2024]. "related the phases of surface-code QEC for coherent and incoherent errors to entanglement phases." [Behrends2024] follow up with FLO.
"The central idea is to decompose (possibly non-Clifford) noise channels into the sum of completely stabilizer preserving (CSP) channels. We simulate the circuits by sampling CSP channels according to quasiprobability distributions, which are obtained from the decompositions." [Hakkaku2021]
Approximate with Depolarizing Approach
Pauli Twirling may be used to turn coherent noise into depolarizing ones. Although the gate implementing Pauli Twirling may also have coherent error. This makes the error channel after twirling not exactly a simple depolarizing channel. A good estimation of such noise channel, original coherent nosie + coherent noise in Pauli Twirling, into mixed depolarizing channel was proposed by [magesan2013]. They simulate with monte carlo sampling. This work has a follow up, not sure what's the contribution [puzzuoli2014]. A following work expands the noise model, including random insertion of one-qubit gate and measurement, which is simulatable under this framekwork [gutierrez2013]. A follow up was done in [Tomita2014] for different code and different noise model.
Deviation of coherent error from Pauli Channel on surface code was studied [bravyi2018]. The noise threshold was accurate with the approximation but the logical error was under-estimated.
Measurement decoheres coherent error and can model as Pauli noise.[beale2018] "after perfect syndrome measurement, the syndrome averaged logical off-diagonal terms of the error channel decays exponentially with respect to the code distance, and the decay is faster than that from the logical diagonal terms. Then, they made a conclusion that the syndrome measurements of the stabilizer code decohere independent coherent errors." [zhao2021]
Analytic Study
Development in this regime was largely due to the following method.
Logical Qubit Dyanmics Approach
To approximate threshold for concatenated code, an effective quantum channel acting on the logical qubit(s) was used to study the dynamics of diagonal noise, each physical qubit under symmetric depolarizing channel, acting on all physical qubits [rahn2002]. This approach was later generalized to working for arbitrary noise in [fern2006]. We know it as the "Pauli Transfer Matrix"[iverson2020]. "However, this technique is not applicable to topological codes, which are more feasible in practical experiments"[Suzuki2017] Cannot approximate non-Clifford noise with Clifford channel which guarantees efficient simulation. Furthermore, the threshold is higher than acutal one due to ignoring encoding and decoding error.
Progress in this subsection is mainly due to the 2002 paper [rahn2002]. In 2016, focusing on how noise scales with code distance and concatenation level, work for repeition code to analytically analyze the amount of coherent error vs incoherent error on the level of logic qubit. [greenbaum2017]
In 2019, logical qubit channel was studied under many five-qubit, Steane, Shor, surface code but measurement was not considered.[huang2019]
In 2020, Obtaining different error syndrom by measurement may changes the effective logical channel. Studying the degeneracy collects equivalent logical channel hence improves simualtion [beal2020].
In 2020, analyzed toric code. "The coherence of the logical channel becomes strongly suppressed as the block length of the quantum error-correcting code increases, assuming that the noise is sufficiently weak and sufficiently weakly correlated." "To decode, one measures the error syndrome, and then applies a recovery operation conditioned on the syndrome. For a large code, many different syndromes are possible, and only the errors which are projected onto the same syndrome value can interfere constructively, while errors projected onto different syndrome values add stochastically."[iverson2020]
In 2023, Answers the question "how dose the coherency of the noise channel affect the logical diagonal terms in (Pauli Transfer Matrix) and the success probability of error corrections."[zhao2021] I.e, the toric code becomes an approximate quantum error correction which recovers the logical state with $\epsilon$ infidelity with the true logic state[zhao2021].
In 2023, five-qubit code performance including measurment, decoding was studied [liu2023]. I am not clear what they meant in [Wagner2023] about each syndrom correspond to a different logical map. However, we are not sure if this limitation manifests itself on measurement free schemes.
Experimental paper that characterizes error model with randomized compiling [hashim2022].
<!– how large of code can we simulate: how large how accurate in general methods approach –> What kind of specific noise model, what is the motivation
TensorNetwork: 153 physical qubit, $\mathcal{2^{\sqrt{n}}}$ for $n$ physical qubit. [Darmawan2017]
Matchgate, FLO: 2401 qubits, code distance 49, surface code. [bravyi2018]
Analytic Method: they obtained a scaling. [iverson2020]
Approximate with Pauli Noise is code specific and they don't relate to code size directly. (In what situation can we do this approximation)
Why don't people talk about size in statistical mapping method?
Consider like our case, only single qubit gate error, but each gate has different error.
"Unlike probabilities of incoherent errors, quantum amplitudes now yield complex Boltzmann weights." [venn2023] What weight does incoherent error give.
<!– is coherent error similar to incoherent error upto a constant in statistical limit –>
In the statistical mapping language, coherent noise causes the system to carry out an insulator-metal transition while incoherent error causes the network to carry out an insulator-insulator transition when noise goes up. [venn2023] Furthermore, characterizing the amount of coherent error requires a different approach than the currently employed randomized benchmarking [kueng2016].
Does Preskill's work support this? What does the Brown paper say about this in terms of measurement?
On the other hand, for surface code at least,
Quotes
"toric code subject to such independent coherent noise, and for minimal- weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance" [iverson2020].
"our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case" [iverson2020].
Questions
Do people concatenate surface code? Why did [Dennis2001] make a fuss about the sufrace code and concatenated code?
- beale2018(Beale et al., 2018)
- marton2023(Marton and Asboth, 2023)
- rahn2002(Rahn et al., 2002)
- fern2006(Fern et al., 2006)
- Tomita2014(Tomita and Svore, 2014)
- magesan2013(Magesan et al., 2013)
- puzzuoli2014(Puzzuoli et al., 2014)
- gutierrez2013(Gutiérrez et al., 2013)
- jouzdani2014(Jouzdani et al., 2014)
- Darmawan2017(Darmawan and Poulin, 2017)
- Suzuki2017(Suzuki et al., 2017)
- greenbaum2017(Greenbaum and Dutton, 2017)
- bravyi2018(Bravyi et al., 2018)
- iverson2020(Iverson and Preskill, 2020)
- venn2020(Venn and Béri, 2020)
- Hakkaku2021(Hakkaku et al., 2021)
- zhao2021(Zhao and Liu, 2021)
- venn2023(Venn et al., 2023)
- hashim2022(Hashim et al., 2022)
- pataki2024(Pataki et al., 2024)
- Behrends2024(Behrends et al., 2024)
- huang2019(Huang et al., 2019)
- aliferis2005(Aliferis et al., 2005)
- beal2020(Beale and Wallman, 2020)
- Wagner2023(Wagner et al., 2023)
- liu2023(Liu, 2023)
- Dennis2001(Dennis et al., 2001)
- Wang2002(Wang et al., 2002)
- Katzgraber2009(Katzgraber et al., 2009)
- Chubb2018(Chubb and Flammia, 2018)
- kueng2016(Kueng et al., 2016)