Quantum computation and computational issues related to quantum mechanics

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Continued Fraction Algorithm in Shor's Algorithm

I am just trying to make the final link of Shor's algorithm clear. Here $r$ is the order of $x$ modulo $N$. We have a number $\psi$, which for a rational number $\dfrac{s}{r}$ satisfies ...
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Which n-qubit Hamiltonians are measurable in poly-time?

Which n-qubit Hamiltonians (or, equivalently Hermitian operators) can feasibly be measured in polynomial time? (as opposed to simulated in poly-time) What the relationship to the complexity of the ...
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Proofs to verify quantum states without revealing their description

Consider the following function $$f_s: k \rightarrow \lvert \psi_k \rangle$$ where $s,k$ are bit strings, and $\lvert \psi_k \rangle$ is a $n$-qubit state. Assume the function is a one-to-one ...
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Hardness of finding similar (quantum) states

Consider a quantum state $\lvert \psi \rangle$, we know from the no cloning theorem, that it cannot be perfectly cloned. Also, loosely speaking, that it can be imperfectly cloned s.t. one can produce ...
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How well can an arbitrary (unknown) quantum state be imperfectly cloned?

How well can an arbitrary unknown (quantum) state $\rvert \psi \rangle = \alpha\rvert 0 \rangle + \beta \rvert 1 \rangle$, be imperfectly/approximately cloned? Given an unknown state ${\rvert \psi ...
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Quantum multi valued decision diagrams [migrated]

I recently came across the paper "QMDD: A Decision Diagram Structure for Reversible and Quantum Circuits" by Thornton and Miller. It deals with a way of compactly representing transformation matrix ...
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Clarification for argument in proof of search in N^1/3 queries with hidden variables/non-collapsing measurements

Let $N=2^n$. In Aaronson's Quantum Computing and Hidden Variables (1) and the recent follow up by Aaronson, Bouland, Fitzsimons, and Lee The space "just above" BQP (2), we consider models of ...
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Probabilistic and quantum analog of $FP$ and $FNP$?

Is there any analog of the computational classes $FP$ and $FNP$ with probabilistic or quantum Turing machines? If so, what are the relation with other computational classes?
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If P = NP were true, would quantum computers be useful?

Suppose that P = NP is true. Would there then be any practical application to building a quantum computer such as solving certain problems faster, or would any such improvement be irrelevant based on ...
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Efficient generation of permutational invariant quantum states

Starting from $|00\cdots 0\rangle$, can permutational invariant quantum states, i.e. the following one: $$ |\psi_n\rangle = \frac1{n!} \sum \prod_{\pi\in S_n} ...
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How does one extend local checkability to quantum complexity classes?

How does one extend local checkability to quantum complexity classes like BQP? Or are quantum algortihms inherently holistic?
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Quantum algorithms for QED computations related to the fine structure constants

My question is about quantum algorithms for QED (quantum electrodynamics) computations related to the fine structure constants. Such computations (as explained to me) amounts to computing Taylor-like ...
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The relationship between QCMA and QMA in the Turing and Communication model

First my background about computational complexity is still beginner. Recent paper published by Klauck and Podder [KP14] show that for the first time an exponential gap between computing partial ...
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Is there a finite unitary gate set which can exactly realise all QFTs of order $2^n$?

I am considering ideas about exact quantum algorithms. In particular, I am considering likely limitations of $\mathsf{EQP}$, which consists of languages exactly decideable by polytime-uniform quantum ...
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Why are sub-normalized states studied in quantum computation?

By basic postulates of QM, any state of a system is described by a normalised density operator. Now i fail to see why people study sub-normalized states ( e.g.: In generalised fidelity etc). I'd be ...
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Why is shifting bits different from shifting qubits?

In classical circuit complexity, shifting bits is considered gratis; all you have to do is reorganizing wires between corresponding gates. By contrast, shifting qubits is typically done by using a ...
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What is the Quantum Cheshire Cat experiments' import to Quantum Computing?

What is the significance of the Quantum Cheshire Cat to Quantum Computing? To recap, the Quantum Cheshire Cat experiment proved it was possible to separate a neutrons' spin from the neutron. Or, in ...
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Complexity class for quantum computer with commutative gates

BQP is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. In quantum computer allowed operations can be ...
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Was the reduction in Shor's algorithm originally discovered by Shor?

This is a "historical question" more than it is a research question, but was the classical reduction to order-finding in Shor's algorithm for factorization initially discovered by Peter Shor, or was ...
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Factoring as a decision problem

I've seen in multiple places stating that factoring is in BQP and referencing Shor's algorithm, but Shor's algorithm is not solving a decision problem. How can factoring be restated in a decision ...
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Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...
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Are there any cases where quantum has given insight for classical algorithms?

To be more specific, has it ever happened that we've made some kind of significant improvement to a classical algorithm or problem as a result of some "trick" or insight gained from looking at quantum ...
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Is any QMA-intermediate problem known?

Similar to the class of classical NP-intermediate problems (e.g. Graph Isomorphism), is there any "QMA-intermediate" problem known, that is in QMA but not known to be QMA-complete? Has this been ...
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Known time complexity advantage of quantum algorithms over classical algorithms [duplicate]

I know that this question may depend on how one formulates each complexity class, but in general, what time complexity advantage does quantum algorithms have over classical algorithms?
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Applications of HHL's algorithm for solving linear equations

In HHL's algorithm for solving a system of linear equations (HHL = Harrow, Hassidim and Lloyd) the output is a quantum state rather than explicit information. Has anyone been able to apply knowledge ...
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The randomized query complexity of the conjoined trees problem

An important 2003 paper by Childs et al. introduced the "conjoined trees problem": a problem admitting an exponential quantum speedup that's unlike just about any other such problem that we know of. ...
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Explicit error bounds on the abelian hidden subgroup problem

What are some explicit forms for the error probability in the typical quantum abelian hidden subgroup algorithm as a function of oracles queries? Ettinger, Hoyer, and Knill give a result that the ...
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Quantum algorithms based on transforms other than Fourier transforms

In Quantum Computation and Quantum Information by Nielsen and Chuang they say that many of the algorithms based on quantum Fourier transforms rely on the Coset Invariance property of Fourier ...
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Oracular separations between poly- and log-depth quantum circuits

The following problem appears in Aaronson's list Ten Semi-Grand Challenges for Quantum Computing Theory. Is $\mathsf{BQP}=\mathsf{BPP}^{\mathsf{BQNC}}$ In other words, can the "quantum" part of ...
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Non-Transversal Fault Tolerant Gate

I have always heard that transversality is a sufficient, not a necessary condition for fault-tolerance in quantum computation. However, I have never seen any examples of non-transversal fault tolerant ...
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Is there a formal proof that quantum computing is or will be faster than classical computing?

Rather than empirical evidence, by what formal principles have we proved that quantum computing will be faster than traditional/classical computing?
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Quantum Fourier Transform question regarding measurement

When we use the quantum fourier transform, for a function, the output is entangled, so if a measurement is made on the output, the result may not be that of the function that one wanted in the first ...
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Why is BQPSPACE in PSPACE if it can have doubly exponentially long running times?

The standard proof that BQPSPACE is in PSPACE relies on a Savitch game type analysis on path integrals. However, it assumes the time running length for BQPSPACE is at most exponentially long. This is ...
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How the errors of the measured quantities of an adiabatic Hamiltonian are inversely proportional to the square root of the number of measurements?

I am going through the paper, Solving the graph-isomorphism problem with a quantum annealer, by Hen et. al. In the last line of the second paragraph of the second column of page 2, it says, Since ...
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A quantum algorithm for GCD

Does anyone know of a direct quantum algorithm for computing GCD, - There could be quantum gates for addition subtraction constructed explicitly, using CNOT, etc. - the construction can be done in ...
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Hardness of quantum circuit equivalence?

Given two poly-sized quantum circuits $C_1$ and $C_2$ on $n$ qubits with a universal gate set generated by some finite set of one and two qubit gates. I'm thinking of the gates $\langle H, T, ...
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What complexity issues are there in considering quantum algorithms with infinite gate-sets?

Short Version Suppose that you want to consider a model of quantum computation in which the gates used in the circuits may depend on the input size. Are there pitfalls to avoid when defining the ...
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Are two-qubit unitaries necessary for universal quantum computation?

I was going through Fast Universal Quantum Computation with Railroad-switch Local Hamiltonians by Daniel Nagaj. In the first sentence of the fifth paragraph on the fourth page, he said, Two-qubit ...
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Generating quadratic optimization problems amenable to quantum annealing

Some context: there is a current debate in adiabatic quantum computing over whether a particular machine, the D-Wave quantum annealer, can outperform a classical algorithm [*]. Earlier this year, a ...
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How powerful is exact “quantum” computing if you suspend unitarity?

Short Question. What is the computational power of "quantum" circuits, if we allow non-unitary (but still invertible) gates, and require the output to give the correct answer with certainty? This ...
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Applications of Quantum Walks?

Can someone explain to me what real world applications could potentially benefit from the study of quantum random walks? I have researched a fair amount on how quantum walks operate and their ...
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Grover's search algorithm for 3 coloring

According to Arora & Barak (pdf), pg. 186, for a polynomial-time computable function $f: \{0,1\}^n \to \{0,1\}$ (represented as a circuit computing $f$), Grover's algorithm finds in ...
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Running Simon's algorithm on D-wave machine

I was wondering whether Simon's algorithm could be run on a D-wave machine. The Simon's algorithm is a promise problem. On the other hand the D-wave machine can run only quadratic unconstrained ...
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How hard it is to approximate the ground state of the (2-D) Hubbard model

The Hubbard model (see also the wikipedea article on the Bose-Hubbard model) is a basic quantum model of solid-state physics. Question: What is the computational complexity of approximating the ...
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Are NQP and QMA comparable?

Both definitions try to create a quantum analog for NP. NQP's definition comes from non-deterministic algorithms: it contains languages for which a Quantum Algorithm accepts with non-zero probability ...
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Why is finding the ground state of a Hamiltonian in QMA?

Why is finding the ground stte of a Hamiltonian in QMA? It's in QMA to figure out if a hamiltonian has any energy eigenvalue within a certain window range which is at least inverse polynomial in ...
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Why is Shor's algorithm in $BPP^{BQNC}$ when needing to uncompute subprocedure call?

Why is Shor's algorithm in $BPP^{BQNC}$? It's true the quantum Fourier transform is in $BPP^{BQNC}$, but the algorithm needs to call a number theoretic function f which has period p which is a factor ...
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Determining if a function is constant or not using period finding

Consider an arbitrary boolean function $$f: {\lbrace 0,1 \rbrace}^n \rightarrow \lbrace 0,1 \rbrace$$ which we write as: $$f(x_1, x_2 ... x_n) $$ where each $x_i$ is a boolean variable We note ...
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Computing the period of a function using a quantum computer

Consider a blackbox function $$f(x): Z \rightarrow \lbrace 0,1 \rbrace $$ Which inputs an integer and outputs 0 or 1 with bit complexity n. If the period P of this function satisfies $$P \in ...
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Period of a Multivariable Function using Quantum Computing

consider a function $$f(x_1,x_2...x_n)$$ I am told it is possible to compute the period of the function as a vector $$<l_1,l_2...l_n>$$ where each l denotes the period of the function for ...