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Theoretical issues related to the quantum treatment of information

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1answer
171 views

Fast classical simulation of quantum algorithms

Are there examples of cases where the classical simulation of a quantum algorithm for a problem outperforms the best previously known classical algorithm for this problem? "Outperforms" doesn't have ...
1
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1answer
63 views

Quantum circuit simulation divergence in results

I'm learning about quantum computing in order to code a simulator. I tried the following circuit in Quirk And ran the same circuit using OPENQASM 2.0: Notice that the input is |11> in both cases, ...
9
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0answers
200 views

How would proof of the Lindelöf hypothesis improve our understanding of computational complexity classes?

A recent press release from the Viterbi School of Engineering at USC discussed the proof of the Lindelöf hypothesis by Athanassios Fokas, a visiting professor from the Department of Applied ...
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1answer
129 views

Complement for joint POVMs?

I'm trying to relate some notions of set theory to POVMs. I firstly explain the scenario with set theory and then in the POVM setting. For some finite $N \in \mathbb{N}$, let $A_i$ and $B_i$ for $i=1,...
2
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1answer
68 views

Computing 'Robustness of Magic' of $n$-bit W states

Question What is the asymptotic robustness-of-magic of a $W$ state over $n$ qubits. Is it $\Theta(n)$? $\Omega(\sqrt{n})$? $O\left(\frac{n}{\lg n} \right)$? Background $W$ states are entangled ...
12
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1answer
998 views

Entropy and computational complexity

There are researcher showing that erasing bit has to consume energy, now is there any research done on the average consumption of energy of algorithm with computational complexity $F(n)$? I guess, ...
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0answers
48 views

Given $f \in \mathbb{Z}_2[x_1, …, x_n]$, are there properties of $f$ which can be used to bound $\mathbb{E}_{i \in [n]}|gap(f) - gap(f + x_i)$|?

Let $f \in \mathbb{Z}_2[x_1, ..., x_n]$ be arbitrary. We define $gap(f) = |f^{-1}(0)| - |f^{-1}(1)|$. I want to understand, as a function of any properties of $f$ you find relevant (number of terms, ...
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0answers
59 views

Noisy channel coding theorem in Quantum information

Why Shannon's noisy channel coding theorem can't be used for Quantum communication applications? Schumacher proved the first Noiseless theorem and there are quantum error correction mechanisms out ...
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0answers
25 views

Are there wholistic models of the universe in terms of Quantum Complexity?

Quantum Computers are an abstraction (a finite circuit of matrices + measurements) that captures the computability properties of local quantum devices. But is there a notion, akin to "computability", ...
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0answers
62 views

Query complexity of quantum search with measuring oracle

Consider the following problem: Let $x\in X$ be a uniformly random value. Let $O$ be an oracle that measures whether the register $Q$ contains $x$. More precisely, $O$ measures $Q$ using the ...
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0answers
52 views

What is the average sensitivity of a quantum circuit with depth $d$ and size $s$?

We have some quantum circuit $C$ with $k$ ancillae and $n$ input bits of depth $d$ and size $s$, and we can define a function $f$ which, for any $x \in \{0, 1\}^n$, is the random variable which is the ...
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3answers
235 views

How can I get $\sum_n e^{i a_n} |n\rangle$ from $\sum_n a_n |n\rangle$?

Suppose that I have a normalized quantum state $\sum_n a_n |n\rangle$, is there a quantum operation/circuit so that I can get $\frac{1}{N} \sum_n e^{i a_n} |n\rangle$ at output? How?
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0answers
101 views

Connection between diamond norm and output purity norm

Setting of the problem: Given a quantum channel $\mathcal{E}: \mathcal{H}_A\rightarrow \mathcal{H}_B$ (where $\mathcal{H}$ refers to a Hilbert space and subscript refers to the quantum register ...
0
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0answers
126 views

Grover's for QUBO form?

My preferred formulation of the quadratic unconstrained binary optimization problem (QUBO) is the following: Find $\min(z)=x'Qx$, where $x$ is a $n$-vector of binary variables, $x'$ is its transpose, ...
7
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1answer
2k views

Oracle Construction for Grover's Algorithm

In Mike and Ike's "Quantum Computation and Quantum Information", Grover's algorithm is explained in great detail. However, in the book, and in all explanations I have found online for Grover's ...
2
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1answer
197 views

Simulating quantum measurements by unitaries

I have seen many papers in which quantum measurements are assumed to be replaced by unitaries. See this quotation from [KW00] for instance: Often we will describe quantum circuits in a high-level ...
1
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1answer
72 views

Finding a basis for quantum measurement with maximum distinguishability

I wish to find a basis state for the quantum measurement of two states which provides the maximum possible distinguishability. In this example let's say we wish to find the best basis ($|\psi\rangle$) ...
9
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1answer
181 views

Where can I find examples of error correcting codes of the following types?

First, apologies if this question is in appropriate or trivial for this site. I'm a physicist looking for some help outside his comfort zone. In PRL 87 167902 (2001) it is claimed that "...for an ...
2
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1answer
139 views

Is graph automorphism Karp-reducible to graph isomorphism under hidden subgroup representation?

The classical representations of the graph automorphism problem is Karp-reducible to the classical representation of the graph isomorphism problem. The sketch of proof for this can be written as ...
1
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0answers
64 views

How to simulate the quantum measurement of a quantum state in Quantum Image

I'm trying to implement (simulate) the Novel Enhanced Quantum Representation (NEQR), which is one of the quantum image representation models, but i'm stuck in the measurement part. In other words i ...
3
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0answers
92 views

Cutting edge of quantum error correction

Often I find myself needing to know the best error correcting code for a certain quantum scenario. For example, suppose my logical systems are 3-dimensional; then what's the most efficient encoding to ...
4
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0answers
94 views

Why primitive rotation is $53.13^\circ$ in the quantum Turing machine used by Vitanyi for Quantum Kolmogrov Complexity?

Right now I am going through Quantum Kolmogorov Complexity Based on Classical Descriptions by Vitanyi. In the introduction, the author assumed the primitive rotation $\theta = 53.13^\circ$ to have ...
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0answers
88 views

Two definitions of $QMA$

In this question, I am trying to understand the equivalence between the following two definitions of the complexity class QMA. In Quantum Computational Complexity, John Watrous defines the class QMA ...
2
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1answer
84 views

Why is it impossible to work with polylog length encoding schemes for quantum circuits?

I am going through Quantum Computational Complexity by John Watrous. On page $12$, he said: The encoding disallows compression: it is not possible to work with encoding schemes that allow for ...
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1answer
766 views

Addition on a quantum computer

From reading https://arxiv.org/pdf/quant-ph/0008033v1.pdf 3n qubits are required to add two n bit numbers. For a simple arithmetic operation such as a+b+c+d where ...
4
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2answers
299 views

When is a non-unitary quantum system only theoretical?

Suppose we construct a non-unitary quantum system α in hilbert space. It entails that this system would have no direct parallel in quantum circuitry as it is a requirement that all quantum gates ...
0
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1answer
83 views

How are new probabilities computed when simulating measurement on a set of qubits?

Suppose I have a set of 3 qubits and I have the probabilities for their distribution. This could be arbitrarily entangled or pure: |000> -> a |001> -> b |010> -> c |011> -> d |100> -> e |101> -> f |...
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1answer
48 views

What are the general classes of measured systems

Imagine there is a class of system such that a measurement can be performed on an exemplar of this class, each measurement producing 1 bit of information. There are no limitation on how many times the ...
8
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1answer
827 views

States and Probability distributions that the 5-qubits IBM computer can produce

Update (January 2018) A new very interesting paper with various experiments on the IBM machine is Five Experimental Tests on the 5-Qubit IBM Quantum Computer by Diego García-Martín, Germán Sierra. ...
1
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1answer
61 views

Can a measurement commuting with the stabilizer of a state disturb the state?

In Nielsen and Chuang's "Quantum Computation and Quantum Information", Section 10.5.3, the authors claim the following: With a system in state $|\psi\rangle$ with stabilizer $g_1,...,g_n$, if a ...
4
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1answer
275 views

theorems for universal set of quantum gates for SU(d)

It seems that there is a theorem that for prime dimension d, the set of Clifford gates and one non-Clifford gate together forms a universal set of quantum gates for SU(d). It also seems that for a ...
4
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1answer
66 views

Is it possible to MAC a quantum state with a classical key under reasonable assumption?

Assume that classical one-way functions secure against quantum adversaries exist. Is it possible, given a quantum state $Q$ and classical secret key $k$, produce a quantum state $AuthQ$ such that: ...
2
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1answer
87 views

Is it possible to encrypt quantum states under reasonable assumptions?

Is it possible to encrypt a quantum state, such that a $BQP$ attacker who does not know the secret key cannot obtain any information about the original state, but a $BQP$ decryptor with the key can ...
3
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1answer
158 views

Quantum GCD circuit: On reversibility and clearing ancillae

Originally posted on PHYS, however, obviously it has more to do with CS I am currently trying to implement a circuit for computing the greatest common divisor in the Quantum Computing Language. In my ...
1
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1answer
59 views

How the hardness of hidden subgroup problem in $S_n$ changes as the order of the subgroup grows?

In Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. In this paper it is showed that no hidden subgroup algorithm can distinguish the trivial ...
3
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2answers
113 views

Is there any hidden subgroup of a symmetric group which can be efficiently determined?

There have been a number of cases where efficient hidden subgroup algorithms have been found for specific non-Abelian groups with very specific structures. Why haven't we found any efficient quantum ...
3
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0answers
83 views

Why hidden subgroup problem is easy for very large subgroup?

I am going through QUANTUM MECHANICAL ALGORITHMS FOR THE NONABELIAN HIDDEN SUBGROUP PROBLEM by Grigni et al. On page 2, it is said that solving the hidden subgroup problem becomes very easy when the ...
0
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2answers
109 views

Entropy inequality of joint von Neumann entropy given some marginal inequalities

Assume that I have a pure, multipartite state $\omega_{ABCD}$ and a unitary $U(\omega_{ABCD})=\tau_{ABCD}$. The effect of $U$ on $\omega$ results in $$ S(\omega_B)<S(\tau_B) $$ and $$ S(\omega_A)&...
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1answer
70 views

Question about discarding the second register in the standard approach of hidden subgroup algorithm

My questions: What does discarding the second register mean for the standard approach of hidden subgroup algorithm? Why does discarding let the first register end up in a mixed state? My ...
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1answer
133 views

Dimension of the Fourier transform for $S_5$ [closed]

My question: What is the dimension of the Fourier transform for $S_5$? My effort: The dimensions of the seven irreps of $S_5$ are $1,1,4,4,5,5,6$. According to the notes of Andrew Childs, the ...
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0answers
52 views

First register in the hidden subgroup representations of Simon's and graph isomorphism problems

The Simon's problem involves a function which takes binary strings as inputs. One seeks to find the period of the function which acts on those inputs. In the standard method, the first register has ...
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1answer
212 views

Generating “infinite” randomness from a constant number of sources

I recently came across a paper by Coudron and Yuen on randomness expansion using quantum devices. The main result of the work is that it is possible to generate "infinite" randomness from a constant ...
3
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0answers
59 views

Does simulating chiral gauge theories lie within BQP?

In theoretical physics, there is a branch of quantum field theory dealing with chiral gauge theories. It has been conjectured by Feynman [1] and others that all quantum field theories can be simulated ...
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1answer
237 views

How does one determine if a mixed bipartite quantum state is entangled or not?

My question is based on the structure of the NP-hardness proof in section 6 (page 17) of this paper, http://arxiv.org/pdf/quant-ph/0303055v1.pdf Mathematically one can think of being given a positive ...
5
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1answer
128 views

Shor's quantum error correction code with unknown basis

$\newcommand{\ket}[1]{\lvert #1 \rangle}$I've met a problem in quantum secret sharing which involves the use of a quantum error-correction code. (let's make it simple to be the 9-qubit Shor code) In ...
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1answer
3k views

Travelling sales man with Quantum Computers [closed]

I know that it takes billions of years to solve the travelling sales man when n = 25 (Number of cities). I am wondering how fast can a quantum computer solve the travelling sales man problem (for ...
14
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5answers
784 views

The utility of Renyi entropies?

Most of us are familiar with — or at least have heard of — the Shannon entropy of a random variable, $H(X) = -\mathbb{E} \bigl[ \log p(X)\bigr]$, and all the related information-theoretic ...
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0answers
92 views

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 mapping....
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0answers
29 views

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 ...
3
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1answer
148 views

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 \...