## 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

### OpenURL

### 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.

### Citations

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Citation Context ...the minimum k for which |ψn〉 can be written as k i=1 αi � � � � � � � � �ϕA i ⊗�ϕB i , where �ϕA i and �ϕB i are states of qubits in A and B respectively. (χA (|ψn〉) is known as the Schmidt rank; see =-=[36]-=- for more information.) Let χ(|ψn〉) = maxA χA (|ψn〉). Then |ψn〉 ∈ Vidal if and only if χ(|ψn〉) ≤ p (n) for some polynomial p. ΨP contains the states |ψn〉 such that for all n and ε > 0, there exists a ... |

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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 ... ∣ ∣α (p(n)) 1y,M,x ≥ 2 3 , ∣ efficiently (as well as Λ(M,x)), we would then have a ΠP 2 predicate expressing that Wx ≥ 2/3. This follows since we can do approximate counting via hashing in AM ⊆ ΠP 2 =-=[26]-=-, and thereby verify that an exponentially large sum of nonnegative terms is at least 2/3, rather than at most 1/3. The one further fact we need is that in our ΠP 2 (∀∃) predicate, we can take the exi... |

181 |
Algebraic Complexity Theory
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Citation Context ...llary 15 does not follow from Raz's results is that polynomial-size formulas for the permanent and determinant are not known; the smallest known formulas for the determinant have size n O(log n) (see =-=[14]-=-). We have shown that not all coset states are tree states, but it is still conceivable that all coset states are extremely well approximated by tree states. Let us now rule out the latter possibility... |

154 |
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Citation Context ...um circuit of size p (n + log (1/#)) that maps the initial state |0## p(n) to a state |## such that ##| # |# n ## |0## (p(n)-n) # # 1 - #, for some polynomial p. Because of the Solovay-Kitaev Theorem =-=[28, 37]-=-, the definition of #P is invariant under the choice of universal gate set. We now formalize the notion of tree size of a quantum state, which will be used throughout this paper. Definition 1 A quantu... |

153 |
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Citation Context ...Given a coset C in Z n 2 , we define the coset state |C# as follows: |C# = 1 # |C| # x#C |x# . Coset states arise as codewords in the class of quantum error-correcting codes known as stabilizer codes =-=[15, 23, 42]-=-. Our interest in these states, however, arises from their large tree size rather than their errorcorrecting properties. For an integer k # 0, let E k,n be the following distribution over cosets C. Ch... |

148 | Number-theoretic constructions of efficient pseudo-random functions
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- 1997
(Show Context)
Citation Context ...sk. Proving superpolynomial formula-size lower bounds for ‘explicit’ functions is a notorious open problem, as it would imply complexity class separations such as NC 1 ̸= P. Indeed, Naor and Reingold =-=[36]-=- gave pseudorandom functions as hard as factoring in NC 1 , so the arguments of Razborov and Rudich [40] imply that any ‘natural’ proof of a superpolynomial lower bound on formula size would yield a s... |

146 |
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(Show Context)
Citation Context ...) is the minimum size of a multilinear formula that represents p. Then given a function f : {0, 1} n # C, we define MFS (f) = min p : p(x)=f(x) #x#{0,1} n MFS (p) . (Actually p turns out to be unique =-=[38]-=-.) We can also define the #-approximate multilinear formula size of f , MFS # (f) = min p : #p-f# 2 2 ## MFS (p) where #p - f# 2 2 = # x#{0,1} n |p (x) - f (x)| 2 . (This metric is closely related to ... |

136 | Multiple particle interference and quantum error correction
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Citation Context ...Given a coset C in Z n 2 , we define the coset state |C# as follows: |C# = 1 # |C| # x#C |x# . Coset states arise as codewords in the class of quantum error-correcting codes known as stabilizer codes =-=[15, 23, 42]-=-. Our interest in these states, however, arises from their large tree size rather than their errorcorrecting properties. For an integer k # 0, let E k,n be the following distribution over cosets C. Ch... |

119 |
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Citation Context ...Raz's method is currently unable to prove exponential lower bounds; n#354 n) is the best it can show. Raz's method is a beautiful combination of the Furst-Saxe-Sipser technique of random restrictions =-=[19]-=-, with matrix rank arguments as used in communication complexity. We now outline the method. Given a function f : {0, 1} n # C, let a k-restriction R (for 0 # k # n/2) set n - 2k of the variables of f... |

113 |
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Citation Context ...proposed that might also explain the evidence. The simplest alternatives are those in which quantum states "spontaneously collapse" with some probability, as in the GRW (Ghirardi-Rimini-Webe=-=r) theory [21]-=-. (Penrose [38] has proposed another such theory, but as mentioned earlier, his suggests that the quantum computing model is too restrictive.) The drawbacks of the GRW theory include violations of ene... |

88 |
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Citation Context ...is described by a wave in three-dimensional phase space, but deny that two particles are described by a wave in six -dimensional phase space. However, the Bell inequality experiments of Aspect et al. =-=[8]-=- and successors have convinced all but a few physicists that quantum entanglement exists, can be maintained over large distances, and cannot be explained by local hidden-variable theories. (3) Schrodi... |

72 |
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Citation Context .... ,xn) as well as constants c1,c2,.... Let Φc denote Φ under a particular assignment c to c1,c2,.... Then a result of Gashkov [22] (see also Turán and Vatan [44]), which follows from Warren’s Theorem =-=[47]-=- in real algebraic geometry, shows that as we range over all c, Φc sign-represents at most ( 2n+4 |Φ| )|Φ| distinct Boolean functions, where |Φ| is the size of Φ. Furthermore, excluding constants, the... |

67 | The Heisenberg representation of quantum computers,” Plenary speech at
- Gottesman
(Show Context)
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... |

54 | The quantum communication complexity of sampling
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- 1998
(Show Context)
Citation Context ...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) ≥ · · · ≥ λN (M) ≥ 0 are... |

53 |
Measurement-based quantum computation on cluster states
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(Show Context)
Citation Context ...der the "cluster states" defined by Briegel and Raussendorf [14]. These states have attracted a great deal of attention because of their application to quantum computing via 1-qubit measurem=-=ents only [40]-=-. For our purposes, a two-dimensional cluster state is an equal superposition over all settings of a # n # n array of bits, with each basis state having a phase of (-1) r , where r is the number of ho... |

47 | Multi-linear formulas for permanent and determinant and of super-polynomial size
- Raz
- 2004
(Show Context)
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... |

45 | Improved simulation of stabilizer circuits
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(Show Context)
Citation Context ... Markov’s inequality, it follows that for at least an 0.04 fraction of S’s, rank � � MS|P = 2n/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 � n2 /log n � . So a corollary of Theorem 12 is that ΨP �⊂ Tree. Since fS clearly has a (non-multilinear) arithmetic f... |

43 |
Separability of very noisy mixed states and implications for NMR quantum computing
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(Show Context)
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 ... |

42 | Adiabatic quantum state generation and statistical zero knowledge
- Aharonov, Ta-Shma
- 2003
(Show Context)
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... |

38 |
Efficient classical simulation of slightly entangled quantum computations
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- 2003
(Show Context)
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... |

36 |
Quantum superposition of distinct macroscopic states
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(Show Context)
Citation Context ...in coherent superposition states. However, recently Arndt et al. [7] have performed the double-slit interference experiment using C 60 molecules (buckyballs) instead of photons; while Friedman et al. =-=[18]-=- have 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. Though sho... |

36 |
The variation of the spectrum of a normal matrix
- Hoffman, Wielandt
- 1953
(Show Context)
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) ≥ · · · ≥ ... |

36 |
Reingold.Number-theoretic construc-tions of efficient pseudo-random functions
- Naor, O
- 2004
(Show Context)
Citation Context ...sk. Proving superpolynomial formula-size lower bounds for `explicit' functions is a notorious open problem, as it would imply complexity class separations such as NC 1 #= P. Indeed, Naor and Reingold =-=[36]-=- gave pseudorandom functions as hard as factoring in NC 1 , so the arguments of Razborov and Rudich [40] imply that any `natural' proof of a superpolynomial lower bound on formula size would yield a s... |

35 |
Nonlinear quantum mechanics implies polynomialtime solution for NP-complete and #P
- Abrams, Lloyd
- 1998
(Show Context)
Citation Context ...d around. How do we know that the successor to quantum mechanics will limit us to BPP, rather than letting us solve (say) PSPACE-complete problems? This is more than a logical point. Abrams and Lloyd =-=[3]-=- show that a wide class of nonlinear variants of the Schrodinger equation would allow NP-complete and even #P-complete problems to be solved in polynomial time. And Penrose [38], who proposed a model ... |

35 |
t Hooft, Quantum gravity as a dissipative deterministic system, Classical and Quantum Gravity 16
- unknown authors
- 1999
(Show Context)
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... |

27 | NP is as easy as detecting unique solutions. Theoret - Valiant, Vazirani - 1986 |

27 |
Persistent entanglement in arrays of interacting particles
- Briegel, Raussendorf
(Show Context)
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... |

25 | Testing the limits of quantum mechanics: motivation, state of play, prospects
- Leggett
(Show Context)
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... |

19 |
Quantum gravity as a dissipative deterministic system
- Hooft
- 1999
(Show Context)
Citation Context .... Finally, as pointed out to us by Rob Spekkens, standard quantum error-correction techniques might be used to overcome GRW-type decoherence. A second class of alternatives includes those of ’t Hooft =-=[43]-=- and Wolfram [48], in which something like a deterministic cellular automaton underlies quantum mechanics. On the basis of his theory, ’t Hooft predicts that “[i]t will never be possible to construct ... |

17 |
An algorithmic benchmark for quantum information processing
- Knill, Laflamme, et al.
(Show Context)
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... |

15 | Counting, fanout and the complexity of quantum ACC
- Green, Homer, et al.
(Show Context)
Citation Context ...e 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 [30]; Vidal [46]; and Green et al. =-=[28]-=-. Here we propose a general framework for the question. For simplicity, we limit ourselves to pure states |ψn〉 ∈ H ⊗n 2 with the fixed orthogonal basis {|x〉 : x ∈ {0,1}n }. Also, by ‘states’ we mean i... |

14 |
Wave-particle duality of C60 molecules
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- 1999
(Show Context)
Citation Context ...s. Accepting two- and three-particle entanglement is not the same as accepting that whole molecules, cats, humans, and galaxies can be in coherent superposition states. However, recently Arndt et al. =-=[7]-=- have performed the double-slit interference experiment using C60 molecules (buckyballs) instead of photons; while Friedman et al. [20] have found evidence that a superconducting current, consisting o... |

13 | Size-depth tradeoffs for Boolean formulae
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- 1994
(Show Context)
Citation Context ...] says that given an arithmetic formula Φ, there exists an equivalent formula of depth O (log |Φ|) and size O (|Φ| c ), where c is a constant. Bshouty, Cleve, and Eberly [14] (see also Bonet and Buss =-=[10]-=-) improved Brent’s theorem to show that c can be taken to be 1 + ε for any ε > 0. So it suffices to show that, for ‘division-free’ formulas, these theorems preserve multilinearity (and in the MOTS cas... |

12 | Size-depth tradeoffs for algebraic formulae
- Bshouty, Cleve, et al.
(Show Context)
Citation Context ...classical theorem of Brent [12] says that given an arithmetic formula Φ, there exists an equivalent formula of depth O (log |Φ|) and size O (|Φ| c ), where c is a constant. Bshouty, Cleve, and Eberly =-=[14]-=- (see also Bonet and Buss [10]) improved Brent’s theorem to show that c can be taken to be 1 + ε for any ε > 0. So it suffices to show that, for ‘division-free’ formulas, these theorems preserve multi... |

11 |
SHPILKA: Learning arithmetic circuits via partial derivatives
- KLIVANS, A
- 2003
(Show Context)
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]... |

11 |
Entangled quantum state of magnetic dipoles
- Ghosh, Rosenbaum, et al.
(Show Context)
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... |

9 | Quantum lower bound for recursive fourier sampling
- Aaronson
(Show Context)
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 ... |

9 |
On the computation of Boolean functions by analog circuits of bounded fan-in (extended abstract
- Turán, Vatan
- 1994
(Show Context)
Citation Context ...takes as input the binary string x = (x1,... ,xn) as well as constants c1,c2,.... Let Φc denote Φ under a particular assignment c to c1,c2,.... Then a result of Gashkov [22] (see also Turán and Vatan =-=[44]-=-), which follows from Warren’s Theorem [47] in real algebraic geometry, shows that as we range over all c, Φc sign-represents at most ( 2n+4 |Φ| )|Φ| distinct Boolean functions, where |Φ| is the size ... |

8 |
Implementation of the five qubit error correction benchmark
- Knill, Laflamme, et al.
(Show Context)
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 |
CP symmetry violation - the search for its origin
- Cronin
- 1980
(Show Context)
Citation Context ...y. Nobel prizes have been awarded in the past ‘merely’ for falsifying a previously held theory, rather than replacing it by a new one. An example is the physics Nobel awarded to Fitch [19] and Cronin =-=[17]-=- in 1980 for discovering CP symmetry violation. Perhaps the key to understanding Levin’s unease about quantum computing lies in his remark that “we have never seen a physical law valid to over a dozen... |

6 |
Polynomial time and extravagant models, in The tale of one-way functions
- Levin
(Show Context)
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 ... |