## Quantum computing, postselection, and probabilistic (2005)

Venue: | Proceedings of the Royal Society A |

Citations: | 38 - 12 self |

### BibTeX

@INPROCEEDINGS{Aaronson05quantumcomputing,,

author = {Scott Aaronson},

title = {Quantum computing, postselection, and probabilistic},

booktitle = {Proceedings of the Royal Society A},

year = {2005},

pages = {0412187}

}

### Years of Citing Articles

### OpenURL

### Abstract

I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.

### Citations

640 | Some optimal inapproximability results
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Citation Context ...ded with a quite 2 A few examples are primality testing in deterministic polynomial time [7], undirected graph connectivity in log-space [30], and inapproximability of the 3-SAT problem unless P = NP =-=[32]-=-. 3 It is clear that PP is closed under complement, so this question is equivalent to asking whether PP is closed under union. 4 Unfortunately, I do not know of any classical result that this helps to... |

477 | Quantum Complexity Theory
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Citation Context ... 3 The Class PostBQP In what follows, I assume basic familiarity with quantum computing, and in particular with the class BQP (Bounded-Error Quantum Polynomial Time) defined by Bernstein and Vazirani =-=[12]-=-. It is now time to define PostBQP more formally. Definition 1 PostBQP is the class of languages L ⊆ {0, 1} ∗ for which there exists a uniform family of polynomial-size quantum circuits {Cn} n≥1 such ... |

222 |
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Citation Context ...ssical computation [4, 9, 25, 29]. What this paper will show is that, even when classical results are already known, quantum computing can sometimes provide new and simpler proofs for them. When Gill =-=[19, 20]-=- defined PP in 1972, he also asked a notorious question that remained open for eighteen years: is PP closed under intersection? 3 In other words, given two probabilistic polynomial-time Turing machine... |

194 | On lattices, learning with errors, random linear codes, and cryptography
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(Show Context)
Citation Context ... proved these results had researchers not long ago “taken aboard” the notion of randomness. 2 Likewise, taking quantum mechanics aboard has already led to some new results about classical computation =-=[4, 9, 25, 29]-=-. What this paper will show is that, even when classical results are already known, quantum computing can sometimes provide new and simpler proofs for them. When Gill [19, 20] defined PP in 1972, he a... |

155 |
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Citation Context ... does seem like an island: change any one aspect, and the whole theory becomes inconsistent or nonsensical. There are many mathematical results supporting this opinion: for example, Gleason’s Theorem =-=[22]-=- and other “derivations” of the |ψ| 2 probability rule [14, 35]; arguments for why amplitudes should be complex numbers, as opposed to (say) real numbers or quaternions [1, 13, 24]; and “absurd” conse... |

153 |
Quantum computations, algorithms and error correction
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Citation Context ...f Shi [31], we can assume without loss of generality that Cn is composed only of Hadamard and Toffoli gates. (This is true even for a postselected quantum circuit, since by the Solovay-Kitaev Theorem =-=[26]-=-, we can achieve the needed accuracy in amplitudes at the cost of a polynomial increase in circuit size.) It is easy to see that NP ⊆ PostBQP. For given a Boolean formula ϕ with n variables, to decide... |

143 |
Undirected ST-connectivity in log-space
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- 2005
(Show Context)
Citation Context ...e attempt failed, and soon thereafter Beigel et al. [10] succeeded with a quite 2 A few examples are primality testing in deterministic polynomial time [7], undirected graph connectivity in log-space =-=[30]-=-, and inapproximability of the 3-SAT problem unless P = NP [32]. 3 It is clear that PP is closed under complement, so this question is equivalent to asking whether PP is closed under union. 4 Unfortun... |

121 | Exponential lower bound for 2-query locally decodable codes via a quantum argument
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(Show Context)
Citation Context ... proved these results had researchers not long ago “taken aboard” the notion of randomness. 2 Likewise, taking quantum mechanics aboard has already led to some new results about classical computation =-=[4, 9, 25, 29]-=-. What this paper will show is that, even when classical results are already known, quantum computing can sometimes provide new and simpler proofs for them. When Gill [19, 20] defined PP in 1972, he a... |

108 | Quantum computability
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Citation Context ...the connection to showing PP closed under intersection. 2 Related Work Besides PostBQP, several other “nondeterministic” versions of BQP have appeared in the literature. Adleman, DeMarrais, and Huang =-=[6]-=- defined NQP to be the class of problems for which there exists a polynomial-time quantum algorithm that accepts with nonzero probability if and only if the answer is ‘yes.’ Then, in a result reminisc... |

94 | Complexity limitations on quantum computation
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Citation Context ... all somehow perverse or degenerate. The Standard Model is not an island, because we do not know of any compelling (non-anthropic) reason why the 6 Another corollary is a result of Fortnow and Rogers =-=[18]-=- that BQP is “low” for PP; that is, PP BQP = PP. For clearly PostBQP BQP = PostBQP. 5 0smasses and coupling constants should have the values they do. Likewise, general relativity is probably not an is... |

86 | PP is closed under intersection
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(Show Context)
Citation Context ... there exist another such machine that accepts with probability greater than 1/2 if and only if A and B both do? The question was finally answered in the affirmative by Beigel, Reingold, and Spielman =-=[11]-=-, who introduced a brilliant technique for representing the logical AND of two majority functions by the sign of a low-degree rational function. Fortnow and Reingold [17] later extended the technique ... |

63 | Succinct quantum proofs for properties of finite groups
- Watrous
- 2000
(Show Context)
Citation Context ...s with nonzero probability if and only if the answer is ‘yes.’ Then, in a result reminiscent of this paper’s, Fenner et al. [15] showed that NQP equals a classical class called coC=P. 4 Also, Watrous =-=[34]-=- defined QMA as the class of problems for which a polynomial-time quantum verifier can be convinced of a ‘yes’ answer by a polynomial-size quantum proof. If we require the proof to be classical, then ... |

61 | Quantum Theory of Probability and Decisions
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- 1999
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Citation Context ...hole theory becomes inconsistent or nonsensical. There are many mathematical results supporting this opinion: for example, Gleason’s Theorem [22] and other “derivations” of the |ψ| 2 probability rule =-=[14, 35]-=-; arguments for why amplitudes should be complex numbers, as opposed to (say) real numbers or quaternions [1, 13, 24]; and “absurd” consequences of allowing nonlinear transformations between states [5... |

52 |
Quantum Theory From Five Reasonable Axioms
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(Show Context)
Citation Context ...example, Gleason’s Theorem [22] and other “derivations” of the |ψ| 2 probability rule [14, 35]; arguments for why amplitudes should be complex numbers, as opposed to (say) real numbers or quaternions =-=[1, 13, 24]-=-; and “absurd” consequences of allowing nonlinear transformations between states [5, 21, 28]. The point of these results is to provide some sort of explanation for why quantum mechanics has the proper... |

48 | Limitations of quantum advice and one-way communication
- AARONSON
- 2005
(Show Context)
Citation Context ...selection has recurred several times in quantum computing. For example, Terhal and DiVincenzo [33] used postselection to show that constant-depth quantum circuits are probably hard to simulate, and I =-=[2]-=- used it to show that BQP/qpoly ⊆ PP/poly (that is, any problem solvable in BQP with polynomial-size quantum advice, is also solvable in PP with polynomial-size classical advice). If we add postselect... |

47 |
the polynomial hierarchy
- Perceptrons
- 1994
(Show Context)
Citation Context ...e Turing machine that makes nonadaptive queries to a PP oracle is no more powerful than PP itself. (Note that a query is “nonadaptive” if it does not depend on the answers to previous queries. Beigel =-=[10]-=- has given evidence that the Fortnow-Reingold result does not generalize to adaptive queries.) Now the class PostBQP is trivially closed under intersection, as well as under polynomial-time truth-tabl... |

47 | The function of
- Wu, Loh, et al.
- 2002
(Show Context)
Citation Context ...under union and intersection is obvious. 2sdifferent approach. This paper could be seen as a vindication of Fortnow et al.’s original approach—all it was missing was quantum mechanics! Admittedly, Li =-=[27]-=- gave another proof along Fortnow et al.’s lines in 1993. However, Li’s proof makes heavy use of rational functions, and seems less intuitive than the one here. 3 The Class PostBQP In what follows, I ... |

44 | Unknown quantum states: The quantum de Finetti representation
- Caves, Fuchs, et al.
- 2002
(Show Context)
Citation Context ...example, Gleason’s Theorem [22] and other “derivations” of the |ψ| 2 probability rule [14, 35]; arguments for why amplitudes should be complex numbers, as opposed to (say) real numbers or quaternions =-=[1, 13, 24]-=-; and “absurd” consequences of allowing nonlinear transformations between states [5, 21, 28]. The point of these results is to provide some sort of explanation for why quantum mechanics has the proper... |

40 | AND NICK REINGOLD: PP is closed under truth-table reductions
- FORTNOW
- 1996
(Show Context)
Citation Context ...eigel, Reingold, and Spielman [11], who introduced a brilliant technique for representing the logical AND of two majority functions by the sign of a low-degree rational function. Fortnow and Reingold =-=[17]-=- later extended the technique to show that PP is closed under “polynomial-time truth-table reductions.” This means that a polynomial-time Turing machine that makes nonadaptive queries to a PP oracle i... |

37 | Threshold computation and cryptographic security
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Citation Context ... advice). If we add postselection to a classical probabilistic polynomial-time Turing machine, then we obtain a complexity class known as BPPpath, which was defined by Han, Hemaspaandra, and Thierauf =-=[23]-=- and which sits somewhere between MA and PP (MA being a probabilistic generalization of NP). Fortnow [16] reports that in 1990, he, Fenner, and Kurtz tried to show PP closed under intersection by (1) ... |

35 |
Nonlinear quantum mechanics implies polynomialtime solution for NP-complete and #P
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- 1998
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Citation Context ...eed to decide in PostBQP whether s < 2n−1 or s ≥ 2n−1 . (As a technicality, we can guarantee � using padding that s > 0.) −n/2 The algorithm is as follows: first prepare the state 2 x∈{0,1} and Lloyd =-=[5]-=-, apply Hadamard gates to all n qubits in the first register and postselect5 on that register being |0〉 ⊗n . This produces the state |0〉 ⊗n |ψ〉 where |ψ〉 = (2n − s) |0〉 + s |1〉 � (2n − s) 2 + s2 . n |... |

34 | Determining acceptance possibility for a quantum computation is hard for the polynomial hierarchy
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- 1999
(Show Context)
Citation Context ...blems for which there exists a polynomial-time quantum algorithm that accepts with nonzero probability if and only if the answer is ‘yes.’ Then, in a result reminiscent of this paper’s, Fenner et al. =-=[15]-=- showed that NQP equals a classical class called coC=P. 4 Also, Watrous [34] defined QMA as the class of problems for which a polynomial-time quantum verifier can be convinced of a ‘yes’ answer by a p... |

32 |
Both Toffoli and controlled-NOT need little help to do universal quantum computation
- Shi
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Citation Context ... second qubit is |1〉 with probability at most 1/3. Here ‘uniform’ means that there exists a classical algorithm that outputs a description of Cn in time polynomial in n. Note that, by a result of Shi =-=[31]-=-, we can assume without loss of generality that Cn is composed only of Hadamard and Toffoli gates. (This is true even for a postselected quantum circuit, since by the Solovay-Kitaev Theorem [26], we c... |

30 | Lower bounds for local search by quantum arguments
- Aaronson
- 2004
(Show Context)
Citation Context ... proved these results had researchers not long ago “taken aboard” the notion of randomness. 2 Likewise, taking quantum mechanics aboard has already led to some new results about classical computation =-=[4, 9, 25, 29]-=-. What this paper will show is that, even when classical results are already known, quantum computing can sometimes provide new and simpler proofs for them. When Gill [19, 20] defined PP in 1972, he a... |

27 |
AND ODED REGEV: Lattice problems in NP intersect coNP
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(Show Context)
Citation Context |

19 |
Weinberg’s non-linear quantum mechanics and superluminal communications
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(Show Context)
Citation Context ...5]; arguments for why amplitudes should be complex numbers, as opposed to (say) real numbers or quaternions [1, 13, 24]; and “absurd” consequences of allowing nonlinear transformations between states =-=[5, 21, 28]-=-. The point of these results is to provide some sort of explanation for why quantum mechanics has the properties it does. In 1998, Abrams and Lloyd [5] suggested that computational complexity could al... |

16 |
Weinberg’s nonlinear quantum mechanics and the Einstein-PodolskyRosen paradox
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(Show Context)
Citation Context ...5]; arguments for why amplitudes should be complex numbers, as opposed to (say) real numbers or quaternions [1, 13, 24]; and “absurd” consequences of allowing nonlinear transformations between states =-=[5, 21, 28]-=-. The point of these results is to provide some sort of explanation for why quantum mechanics has the properties it does. In 1998, Abrams and Lloyd [5] suggested that computational complexity could al... |

14 |
Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games. arXiv e-prints
- Terhal, DiVincenzo
- 2002
(Show Context)
Citation Context ...Naveh [8]. Note that all of these classes are contained in PP [34], and hence in PostBQP. The idea of postselection has recurred several times in quantum computing. For example, Terhal and DiVincenzo =-=[33]-=- used postselection to show that constant-depth quantum circuits are probably hard to simulate, and I [2] used it to show that BQP/qpoly ⊆ PP/poly (that is, any problem solvable in BQP with polynomial... |

11 |
Probabilistic Turing Machines and Complexity of Computations
- Gill
- 1972
(Show Context)
Citation Context ...ssical computation [4, 9, 25, 29]. What this paper will show is that, even when classical results are already known, quantum computing can sometimes provide new and simpler proofs for them. When Gill =-=[19, 20]-=- defined PP in 1972, he also asked a notorious question that remained open for eighteen years: is PP closed under intersection? 3 In other words, given two probabilistic polynomial-time Turing machine... |

9 |
NAVEH: Quantum NP - a survey. quant-ph/0210077
- AHARONOV, T
- 2002
(Show Context)
Citation Context ...verifier can be convinced of a ‘yes’ answer by a polynomial-size quantum proof. If we require the proof to be classical, then we obtain the apparently weaker class QCMA, defined by Aharonov and Naveh =-=[8]-=-. Note that all of these classes are contained in PP [34], and hence in PostBQP. The idea of postselection has recurred several times in quantum computing. For example, Terhal and DiVincenzo [33] used... |

7 | Is quantum mechanics an island in theoryspace
- Aaronson
- 2004
(Show Context)
Citation Context ...ational functions similar to those in [11] could be extracted from the proof of my result.) This paper is based on chapter 15 of my PhD thesis [3]. Some of the results appeared in preliminary form in =-=[1]-=-, before I made the connection to showing PP closed under intersection. 2 Related Work Besides PostBQP, several other “nondeterministic” versions of BQP have appeared in the literature. Adleman, DeMar... |

3 |
My Computational Complexity Web Log
- Fortnow
(Show Context)
Citation Context ...tain a complexity class known as BPPpath, which was defined by Han, Hemaspaandra, and Thierauf [23] and which sits somewhere between MA and PP (MA being a probabilistic generalization of NP). Fortnow =-=[16]-=- reports that in 1990, he, Fenner, and Kurtz tried to show PP closed under intersection by (1) defining a seemingly weaker class, (2) showing that class closed under intersection, and then (3) showing... |

3 |
Environment-assisted invariance, causality, and probabilities in quantum physics
- Zurek
(Show Context)
Citation Context ...hole theory becomes inconsistent or nonsensical. There are many mathematical results supporting this opinion: for example, Gleason’s Theorem [22] and other “derivations” of the |ψ| 2 probability rule =-=[14, 35]-=-; arguments for why amplitudes should be complex numbers, as opposed to (say) real numbers or quaternions [1, 13, 24]; and “absurd” consequences of allowing nonlinear transformations between states [5... |

2 |
Limits on Efficient Computation in the Physical World
- Aaronson
- 2004
(Show Context)
Citation Context ...gh, as pointed out to me by Richard Beigel, a result on rational functions similar to those in [11] could be extracted from the proof of my result.) This paper is based on chapter 15 of my PhD thesis =-=[3]-=-. Some of the results appeared in preliminary form in [1], before I made the connection to showing PP closed under intersection. 2 Related Work Besides PostBQP, several other “nondeterministic” versio... |