## Multilinear Formulas and Skepticism of Quantum Computing (2004)

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Venue: | In Proc. ACM STOC |

Citations: | 29 - 7 self |

### BibTeX

@INPROCEEDINGS{Aaronson04multilinearformulas,

author = {Scott Aaronson},

title = {Multilinear Formulas and Skepticism of Quantum Computing},

booktitle = {In Proc. ACM STOC},

year = {2004},

pages = {118--127},

publisher = {ACM Press}

}

### Years of Citing Articles

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### Abstract

Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural "Sure/Shor separator"---that is, a set of quantum states that can account for all experiments performed to date, but not for Shor's factoring algorithm. We propose as a candidate the set of states expressible by a polynomial number of additions and tensor products. Using a recent lower bound on multilinear formula size due to Raz, we then show that states arising in quantum error-correction require n## additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions. More broadly, we introduce a complexity classification of pure quantum states, and prove many basic facts about this classification. Our goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses that might be experimentally testable in the near future.

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Citation Context ...bjection (A), that quantum computing is impossible for a fundamental physical reason. Although this objection has been raised by several physicists, including Gerard 't Hooft [26] and Stephen Wolfram =-=[47]-=-, we will begin with the arguments of Leonid Levin [33, 34], since those are the best known to computer scientists. The following is a sample of points made by Levin that we were able to understand. W... |

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Citation Context ...sults suggest new experiments that would test candidate Sure/Shor separators more directly. To illustrate, we pose the following challenge to experimentalists. Prepare a `generic' Cli#ord group state =-=[24]-=----that is, a state obtainable by applying Hadamard, CNOT, and #/4 phase rotation gates, starting from |0## n ---on as many qubits as possible. For concreteness, a 9-qubit state stabilized by the foll... |

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Citation Context ...re. Section 5.2 states the conjecture and shows that it implies an n# log n) tree size lower bound for states arising in Shor's algorithm. Our lower bounds use a sophisticated recent technique of Raz =-=[39]-=-, which was introduced to show that the permanent and determinant of a matrix require superpolynomial-size multilinear formulas. Currently, Raz's technique is only able to show lower bounds of the for... |

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Citation Context ...tate # = # |## ##| + (1 - #) 2 -n I 2 n , where I 2 n is the identity (so I 2 n/2 n is the maximally mixed state), and # scales roughly as n/2 n (in current experiments # # 10 -5 ). Braunstein et al. =-=[11]-=- have shown that, if the number n of qubits is less than about 14, then such states cannot be entangled. That is, there exists some representation of # as a mixture of pure states, each of which is a ... |

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Citation Context ...s, which states have polynomial-size classical descriptions of various kinds? This question has been studied from several angles by Aharonov and Ta-Shma [5]; Janzing, Wocjan, and Beth [27]; and Vidal =-=[45]-=-. Here we propose a unified framework for the question. For simplicity, we limit ourselves to pure states |# n # # H# n 2 with the fixed orthogonal basis {|x# : x # {0, 1} n }. Also, by `states' we me... |

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Citation Context ...complexity of quantum states themselves. That is, which states have polynomial-size classical descriptions of various kinds? This question has been studied from several angles by Aharonov and Ta-Shma =-=[5]-=-; Janzing, Wocjan, and Beth [27]; and Vidal [45]. Here we propose a unified framework for the question. For simplicity, we limit ourselves to pure states |# n # # H# n 2 with the fixed orthogonal basi... |

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Citation Context ...llows from the Hoffman-Wielandt inequality. Lemma 14 Let M be an N × N complex matrix, and let IN be the N × N identity matrix. Then ‖M − IN‖ 2 2 ≥ N − rank(M). Proof. The Hoffman-Wielandt inequality =-=[29]-=- (see also [6]) states that for any two N × N matrices M,P, N∑ (σi (M) − σi (P)) 2 ≤ ‖M − P ‖ 2 2 , i=1 where σi (M) is the ith singular value of M (that is, σi (M) = √ λi (M), where λ1 (M) ≥ · · · ≥ ... |

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Citation Context ... This section discusses objection (A), that quantum computing is impossible for a fundamental physical reason. Although this objection has been raised by several physicists, including Gerard 't Hooft =-=[26]-=- and Stephen Wolfram [47], we will begin with the arguments of Leonid Levin [33, 34], since those are the best known to computer scientists. The following is a sample of points made by Levin that we w... |

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Citation Context ...ed, but it is not persistently entangled, since measuring a single qubit in the standard basis destroys all entanglement. By contrast, consider the "cluster states" defined by Briegel and Ra=-=ussendorf [14]-=-. These states have attracted a great deal of attention because of their application to quantum computing via 1-qubit measurements only [40]. For our purposes, a two-dimensional cluster state is an eq... |

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Citation Context ...ave found evidence that a superconducting current, consisting of billions of electrons, can enter a coherent superposition of flowing clockwise around a coil and flowing counterclockwise (see Leggett =-=[34]-=- for a survey of such experiments). Though short of cats, these experiments at least allow us to say the following: if we could build a general-purpose quantum computer with as many components as have... |

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Citation Context ...position |## = 1 4 # (|01# + |10#)# (|010# - |111#) + (|01# - |10#)# (|001# - |100#) - (|00# + |11#)# (|011# + |110#) + (|00# - |11#)# (|000# + |101#) # . Second, Knill, Laflamme, Martinez, and Tseng =-=[31]-=- created a 7-qubit cat state, (|0000000# + |1111111#) / # 2. This has recently been improved to 12 qubits (R. Laflamme, personal communication). Our proposed experiment di#ers in two respects from exp... |

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Citation Context ...gett, there exist systems studied in condensed-matter physics that are strong candidates for having superpolynomial tree size. An example is the magnetic salt LiHo x Y 1-x F 4 studied by Ghosh et al. =-=[24]-=-, which, like the cluster states of Briegel and Raussendorf [14], basically consists of a lattice of spins subject to pairwise nearest-neighbor Hamiltonians. The main di#erences are that the salt latt... |

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Citation Context ...puter that is always in a tree state be simulated classically? The key question seems to be whether the concept class of multilinear formulas is e#ciently learnable. Intriguingly, Klivans and Shpilka =-=[29]-=- gave an e#cient algorithm to learn depth-3 multilinear formulas. However, the quantum simulation algorithm that their result implies turns out to be subsumed by the simulation algorithm of Vidal [45]... |

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Citation Context ...TreeBQP # # P 3 # # P 3 , where TreeBQP is the subclass of BQP corresponding to tree states, and # P 3 # # P 3 is the third level of the polynomial hierarchy PH. 2 By contrast, it is conjectured (see =-=[2]-=-) that BQP ## PH. Section 7 proposes an experiment to test the idea that all states in Nature are tree states. We give evidence that our experiment is feasible with current liquid NMR technology, and ... |

8 |
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Citation Context ...a Sure/Shor separator. As evidence for our experiment's feasibility, we mention two experiments that have already been performed in liquid-state NMR. First, Knill, Laflamme, Martinez, and Negrevergne =-=[30]-=- demonstrated encoding, error detection, and error correction for a code that maps 1 qubit to 5 qubits. This involved preparing nuclear spin states such as the following: |## = 1 4 # |00000# + |10010#... |

7 |
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Citation Context ...ooted in science fiction . . . The present attitude would be analogous to, say, Maxwell selling the Daemon of his famous thought experiment as a path to cheaper electricity from heat. ---Leonid Levin =-=[33]-=- Quantum computing presents a dilemma: is it reasonable to study a type of computer that has never been built, and might never be built in one's lifetime? Some researchers strongly believe the answer ... |

5 |
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Citation Context ...emselves. That is, which states have polynomial-size classical descriptions of various kinds? This question has been studied from several angles by Aharonov and Ta-Shma [5]; Janzing, Wocjan, and Beth =-=[27]-=-; and Vidal [45]. Here we propose a unified framework for the question. For simplicity, we limit ourselves to pure states |# n # # H# n 2 with the fixed orthogonal basis {|x# : x # {0, 1} n }. Also, b... |

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Citation Context ...dicts. Measuring the exact state is unnecessary. Nobel prizes have been awarded in the past `merely' for falsifying a previously held theory, rather than replacing it by a new one. For example, Fitch =-=[17] and -=-Cronin [16] won the physics Nobel in 1980 for discovering that CP symmetry is violated. Perhaps the key to understanding Levin's unease about quantum computing lies in his remark that "we have ne... |

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Citation Context ...probabilities of order 2 -1000 . Should one dismiss such probabilities as unphysical? At the very least, it is not obvious that amplitudes should behave 2 Since this paper was written, Oded Goldreich =-=[25] has also -=-put forward an argument against quantum computing. Compared to Levin's arguments, Goldreich's is easily understood: he believes that Shor states have exponential "nondegeneracy " and therefo... |

1 |
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Citation Context ... usual notions of causality and locality might no longer apply in quantum gravity. As for Wolfram's theory, which involves "long-range threads" to account for Bell inequality violations, we =-=argued in [1]-=- that it fails Wolfram's own desiderata of causal and relativistic invariance. 5 Sure States (already demonstrated) Shor States (suffice for nontrivial factoring) Allowed by local hidden variable theo... |

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1 |
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Citation Context ... the exact state is unnecessary. Nobel prizes have been awarded in the past `merely' for falsifying a previously held theory, rather than replacing it by a new one. For example, Fitch [17] and Cronin =-=[16] won -=-the physics Nobel in 1980 for discovering that CP symmetry is violated. Perhaps the key to understanding Levin's unease about quantum computing lies in his remark that "we have never seen a physi... |

1 |
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Citation Context ...rkov's inequality, it follows that for at least an 0.04 fraction of S's, rank # M S|P # = 2 n/2 for at least an 0.04 fraction of P 's. Theorem 9 then yields the desired result. Aaronson and Gottesman =-=[3]-=- show how to prepare any n-qubit subgroup state using a quantum circuit of size O # n 2 / log n # . So a corollary of Theorem 12 is that #P ## Tree. Since f S clearly has a (non-multilinear) arithmeti... |