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Checking references for intended status: Informational ---------------------------------------------------------------------------- No issues found here. Summary: 0 errors (**), 0 flaws (~~), 1 warning (==), 1 comment (--). Run idnits with the --verbose option for more detailed information about the items above. -------------------------------------------------------------------------------- 2 INTERNET DRAFT M. Ohta 3 draft-ohta-qec-inapplicable-00.txt Tokyo Institute of Technology 4 Intended status: Informational October 30, 2020 5 Expires: May 3, 2021 7 Quantum Error Correction Inapplicable to Really Entangled States 9 Status of this Memo 11 This Internet-Draft is submitted in full conformance with the 12 provisions of BCP 78 and BCP 79. 14 Internet-Drafts are working documents of the Internet Engineering 15 Task Force (IETF). Note that other groups may also distribute working 16 documents as Internet-Drafts. The list of current Internet-Drafts is 17 at http://datatracker.ietf.org/drafts/current. 19 Internet-Drafts are draft documents valid for a maximum of six months 20 and may be updated, replaced, or obsoleted by other documents at any 21 time. It is inappropriate to use Internet-Drafts as reference 22 material or to cite them other than as "work in progress." 24 Copyright Notice 26 Copyright (c) 2020 IETF Trust and the persons identified as the 27 document authors. All rights reserved. 29 This document is subject to BCP 78 and the IETF Trust's Legal 30 Provisions Relating to IETF Documents 31 (http://trustee.ietf.org/license-info) in effect on the date of 32 publication of this document. Please review these documents 33 carefully, as they describe your rights and restrictions with respect 34 to this document. 36 Abstract 38 Though quantum error correction assumes localized error model of Shor 39 that errors on a qubit are caused by interaction with its local 40 environment, enabling essentially classical error correction for 41 unentangled states, the model is applied to entangled states 42 improperly without involving local environment states in the 43 entanglement. 45 That is, when an entangled state (Q) is represented as superposition 46 of unentangled terms (Qi) as Q=Q1+Q2+...+Qn, local environment states 47 around qubits are, in general, different term by term. Q will be, 48 with term-specific error operators (Ei), E1*Q1+E2*Q2+...+En*Qn, not, 49 with a common error operator (E) assumed by Shor, E*(Q1+Q2+...+Qn). 51 A complication is that Shor's error model is a little quantum, 52 allowing for two different local environment states around a qubit. 53 As such, quantum error correction is applicable to some trivially 54 entangled states including states used by Shor code but not to really 55 entangled states. 57 1. Introduction 59 An assumption of noise model for quantum error correction by Shor [1] 60 is "The critical assumption here is that decoherence only affects one 61 qubit of our superposition, while the other qubits remain unchanged. 62 It is not clear how reasonable this assumption is physically, but it 63 corresponds to the assumption in classical information theory of the 64 independence of noise.", which means a qubit suffers from error as a 65 result of interaction with local environment around the qubit but no 66 interaction occurs with other qubits or local environment of other 67 qubits. Though some extension to consider certain interaction 68 between a qubit and other qubits or environment of other qubits is 69 possible, some locality is still assumed. 71 The error model is directly applicable to unentangled, that is, 72 essentially classical, states, resulting in localized errors, 73 corrections of which are essentially classical error correction. 75 However, it is unreasonable to expect such localized errors for 76 entangled states, because the states themselves do not have locality. 77 Actually, with a 2 qubit entangled state: |00>+|11>, if the first 78 qubit coherently interacts with its environment to be |0>, the entire 79 state becomes |00>, which means the second qubit is also affected, 80 Though the case is trivial enough to be explained by Shor's error 81 model as superposition of identity (no error) and sign flip (|0> and 82 |1> become |0> and -|1>, correspondingly) error: 83 |00>=((|00>+|11>)+(|00>-|11>))/2, such an explanation dose not deny 84 lack of locality of errors on entangled states. 86 As Shor overlooked the fact that when qubit states are entangled, 87 their environment states are, in general, also entangled, errors on 88 really entangled states are highly non-local to which quantum error 89 correction is not applicable. 91 That is, when an entangled state (Q) is represented as superposition 92 of (minimum number of) unentangled terms (Qi) as Q=Q1+Q2+...+Qn, 93 local environment states around a qubit are, in general, involved in 94 the entanglement and different term by term, resulting in different 95 error operators (Ei). As a result, Q will be disturbed by noise to be 96 E1*Q1+E2*Q2+...+En*Qn, whereas, Shor thought a common error operator 97 (E) is applicable to all the terms as E*(Q1+Q2+...+Qn). 99 It is obvious that, with some clever encoding using fixed number of 100 extra qubits, effect of E may be compensated, which was quantum error 101 correction, but the extra qubits are not enough to compensate all the 102 Ei's (with quantum algorithms, 'n' will often be exponentially large 103 w.r.t. problem size). 105 A complication is that Shor's error model is a little quantum, 106 allowing for, seemingly despite his intention, two different local 107 environment states around a qubit, which is explained in the next 108 section. 110 2. Why Shor's Error Model is a little Quantum? 112 In [1], Shor explicitly described environment state of a qubit before 113 interaction with the qubit |e0> (same state for |0> and |1>, which 114 should be the intention of Shor) and described interaction 115 (decoherence) process as: 117 |e0>|0> -> |a0>|0>+|a1>|1> 119 |e0>|1> -> |a2>|0>+|a3>|1> 121 where |a0>, |a1>, |a2> and |a3> are environment states after the 122 interaction. |a0>, |a1>, |a2> and |a3> are "not generally orthogonal 123 or normalized" [1] and can be fully independent each other. Ignoring 124 error terms, 126 |e0>|0> -> |a0>|0> 128 |e0>|1> -> |a3>|1> 130 So, if qubit state is |0>, its environment state is |a0>, but, if 131 qubit state is |1>, its environment state us |a3>, different from 132 |a0>. 134 It should also be noted that, as |a0>, |a1>, |a2> and |a3> are fully 135 independent each other, the process may have two different initial 136 environment states as: 138 |e0>|0> -> |a0>|0>+|a1>|1> 140 |e1>|1> -> |a2>|0>+|a3>|1> 142 So, Shor's error model is slightly quantum allowing for different 143 environment states depending on qubit values. 145 As such, errors on trivially entangled states (e.g., superposition of 146 just two unentangled states) such as |00>+|11> and 147 (|000>+|111>)(|000>+|111>)(|000>+|111>) should be correctable. As 148 the latter example is Shor code for |0>, experimental confirmation of 149 Shor's quantum error correction should success, as long as the input 150 qubit to an error correction circuit is unentangled with other qubits 151 outside of the circuit, which is not the case when quantum algorithms 152 are run on quantum computers relying on aggressive entanglement 153 between qubits. 155 It should be noted that, though it does not affect the points of this 156 memo, Shor's representation of qubit and its environment states using 157 tensor product is inappropriate, because, for the interaction, 158 relative phase between them matters (e.g., resulting states of 159 homodyne detection relies on the relative phase), which can be 160 represented by not tensor but Cartesian product. Though |e0>|0> and 161 -|e0>|0> represent a same state, (|e0>, |0>) and (|e0>, -|0>) are 162 different states. 164 It should also be noted that Shor's error model is a little quantum 165 not because sign flip error is quantum specific and classically 166 impossible. It is merely that sign flip error does not occur on 167 modern computers where phase is not used to encode information. In 168 an optical packet router using FDLs (Fiber Delay Lines) as optical 169 buffers (memory), like ancient computers with Mercury delay lines as 170 memory, where QAM (Quadrature Amplitude Modulated) PDM (Polarization 171 Division Multiplexed) signal is sent over the FDLs [2], sign flip 172 errors occur as relative phase errors between polarization modes. 174 3. Conclusions 176 It is shown that not-really-quantum error correction works only for 177 errors with mostly classical locality and is not applicable to non- 178 local errors on really entangled states. 180 As qubit states within quantum computers running quantum algorithms 181 are really entangled, quantum error correction for them is 182 impossible, which makes construction of quantum computers with 183 practical size practically impossible. 185 Entangled states, in general, are a lot noisier than Shor thought, 186 which should be the reason why the states are so fragile easily 187 collapsing to be less noisy less entangled or unentangled states. 189 4. Security Considerations 191 That construction of quantum computers with practical size is 192 practically impossible means quantum computers do not make public key 193 cryptography unsafe, though there may still be some classical 194 algorithm to make it unsafe. 196 5. IANA Considerations 198 This memo has no actions for IANA. 200 Informative References 202 [1] P. W. Shor, "Scheme for reducing decoherence in quantum computer 203 memory", Phys. Rev. A, Oct. 1995, 204 http://www.cs.miami.edu/~burt/learning/Csc670.052/pR2493_1.pdf. 206 [2] M. Ohta, "Optical switching of many wavelength packets: A 207 conservative approach for an energy efficient exascale 208 interconnection network", 2016 IEEE 17th International Conference on 209 High Performance Switching and Routing (HPSR), 210 https://ieeexplore.ieee.org/document/7525641, August 2016. 212 Author's Address 214 Masataka Ohta 215 Tokyo Institute of Technology 216 2-12-1-W8-54, O-okayama, Meguro-ku 217 Tokyo 152-8552 218 JAPAN 220 Phone: +81-3-5734-3299 221 Fax: +81-3-5734-3299 222 EMail: mohta@necom830.hpcl.titech.ac.jp