Parallel "paths" in quantum computing

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Philip Thrift

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Aug 7, 2019, 4:08:50 AM8/7/19
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On Tuesday, August 6, 2019 at 5:29:04 PM UTC-5, Brent wrote:


On 8/6/2019 11:25 AM, Philip Thrift wrote:


On Tuesday, August 6, 2019 at 1:00:23 PM UTC-5, Brent wrote:


On 8/6/2019 6:38 AM, Bruno Marchal wrote:
If the QC does its task effectively, the output basis qbits will be put into definite states,

Relatively to the observer, but in the global state, the observer will inherit the superposition state, by linearity of the tensor products and of the evolution.

In something like Shor's algorithm there is only one final state with non-vanishing probability.  Yet this is the kind of algorithm that Deutsch cites as proving there must be many worlds.

Brent



That there is a multiplicity of somethings  


is the basis for all semantics of quantum computing (by computer scientists) that I have ever seen.

Same for classical computation...there are lots of states or functions.  Did anyone think there had to be multiple worlds for the computer to work?

Brent



There is classical parallel hardware, e.g. made with multiple processors.

Parallelism in quantum computers is achieved by parallel "worlds" or "paths":

Quantum Path Computing

Quantum circuit dynamics via path integrals: Is there a classical action for discrete-time paths?

A “problem of time” in the multiplicative scheme for the n-site hopper
Fay Dowker, Vojtˇech Havlicek, Cyprian Lewandowski, and
Henry Wilkes
"Quantum Measure Theory (QMT*) is an approach to quantum mechanics,
based on the path integral, in which quantum theory is conceived of as a generalized stochastic process." 

The sum-over-histories formulation of quantum computing

@philipthrift

Brent Meeker

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Aug 7, 2019, 2:03:44 PM8/7/19
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On 8/7/2019 1:08 AM, Philip Thrift wrote:
On Tuesday, August 6, 2019 at 5:29:04 PM UTC-5, Brent wrote:


On 8/6/2019 11:25 AM, Philip Thrift wrote:


On Tuesday, August 6, 2019 at 1:00:23 PM UTC-5, Brent wrote:


On 8/6/2019 6:38 AM, Bruno Marchal wrote:
If the QC does its task effectively, the output basis qbits will be put into definite states,

Relatively to the observer, but in the global state, the observer will inherit the superposition state, by linearity of the tensor products and of the evolution.

In something like Shor's algorithm there is only one final state with non-vanishing probability.  Yet this is the kind of algorithm that Deutsch cites as proving there must be many worlds.

Brent



That there is a multiplicity of somethings  


is the basis for all semantics of quantum computing (by computer scientists) that I have ever seen.

Same for classical computation...there are lots of states or functions.  Did anyone think there had to be multiple worlds for the computer to work?

Brent



There is classical parallel hardware, e.g. made with multiple processors.

Parallelism in quantum computers is achieved by parallel "worlds" or "paths":

Quantum Path Computing

Quantum circuit dynamics via path integrals: Is there a classical action for discrete-time paths?

But as you note with scare quotes, calling those "worlds" or  "paths" is just metaphorical.  They are not worlds you can visit or paths you can take.  They are aspects of mathematical abstractions.

Brent


A “problem of time” in the multiplicative scheme for the n-site hopper
Fay Dowker, Vojtˇech Havlicek, Cyprian Lewandowski, and
Henry Wilkes
"Quantum Measure Theory (QMT*) is an approach to quantum mechanics,
based on the path integral, in which quantum theory is conceived of as a generalized stochastic process." 

The sum-over-histories formulation of quantum computing

@philipthrift
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Philip Thrift

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Aug 7, 2019, 2:15:56 PM8/7/19
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If a multiplicity of somethings isn't present in a quantum computer, then how does the speedup occur?

 @philipthrift

Brent Meeker

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Aug 7, 2019, 3:59:04 PM8/7/19
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By not decohering at every bit flip and keeping the single state rotating.

Brent

Philip Thrift

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Aug 7, 2019, 5:56:51 PM8/7/19
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Then how does an answer come out?


Like in the solution  here:  https://arxiv.org/abs/1709.00735

QPC solves specific instances of simultaneous Diophantine approximation problem (NP-hard) as an important application. 

QPC does not explicitly require exponential complexity of resources by combining tensor product space of path histories inherently existing in the physical set-up and path integrals naturally including histories. 


@philipthrift

Lawrence Crowell

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Aug 7, 2019, 7:40:51 PM8/7/19
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On Wednesday, August 7, 2019 at 4:56:51 PM UTC-5, Philip Thrift wrote:

Then how does an answer come out?


By LOCC = local operation and classical communication Largely this is a Hadamard gate for selection and a classical signal on the ancillary states.

LC

Brent Meeker

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Aug 7, 2019, 10:19:57 PM8/7/19
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By decoherence at the end.




Like in the solution  here:  https://arxiv.org/abs/1709.00735

QPC solves specific instances of simultaneous Diophantine approximation problem (NP-hard) as an important application. 

QPC does not explicitly require exponential complexity of resources by combining tensor product space of path histories inherently existing in the physical set-up and path integrals naturally including histories.

Interesting, but looks more aspirational than proven.  It reminds me of attempts to solve NP-hard problems using some analog schemes.

Brent

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Philip Thrift

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Aug 8, 2019, 2:14:26 AM8/8/19
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On Wednesday, August 7, 2019 at 9:19:57 PM UTC-5, Brent wrote:


On 8/7/2019 2:56 PM, Philip Thrift wrote:


On Wednesday, August 7, 2019 at 2:59:04 PM UTC-5, Brent wrote:


On 8/7/2019 11:15 AM, Philip Thrift wrote:

If a multiplicity of somethings isn't present in a quantum computer, then how does the speedup occur?

By not decohering at every bit flip and keeping the single state rotating.

Brent





Then how does an answer come out?

By decoherence at the end.


Brent



Well, sure. But there is still a multiplicity of somethings. 

Histories.


In QMT a system is defined by a triple (Ω, A, D) where Ω is the set of
histories, A is the event algebra and D is the decoherence functional. An event is a set of histories and the event algebra is the set of all events to which the theory assigns a measure. The event algebra is, then, a subset of the power set of Ω. 

The decoherence functional is a function with two arguments D : A × A → C such that
• D(A, B) = D(B, A)
∗ ∀ A, B ∈ A ;
• For any finite collection of events A1, . . . , Am ∈ A, the m × m matrix
Mab := D(Aa, Ab) is positive semi-definite ;
• D(Ω, Ω) = 1 ;
• D(A ∪ B, C) = D(A, C) + D(B, C) ∀ A, B, C ∈ A such that A ∩ B = ∅ .
The quantum measure, µ(E), of an event E ∈ A is given by the diagonal of the decoherence functional D:
µ(E) := D(E, E).

 
@philipthrift


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