Questions tagged [complexity-theory]

For questions regarding complexity analysis of quantum algorithms and comparisons with complexities of classical algorithms.

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What is the explicit best known quantum algorithm for LWE?

Consider the learning with errors(LWE) problem which is known to be hard for quantum computers. Let $q \geq 2$ be a prime integer. Consider us being given (polynomially many samples of) either: $$A, ...
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32 views

What time complexity is considered difficult for quantum computers? [closed]

Not space complexity this time. Just want to know the limitations of its performance
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118 views

What is the relationship between the size of the Hilbert space for boson sampling and the complexity of classical simulating it?

My intuition is that the fastest classical algorithm for simulating some kind of noiseless quantum sampling process should scale roughly with the dimension of the Hilbert space: you would need to ...
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123 views

Has the possibility of there being a classical cryptography algorithm able to withstand quantum computing been proven?

Has it been proven, that a classical codec (encoder-decoder) (classical meaning one that doesn't require a quantum system for its operation) is possible, such that a quantum computer cannot crack it? ...
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Is the exponential speedup and output $\langle x|M|x\rangle$ in contradiction in HHL algorithm?

Isn't the exponential speedup and the output $\langle x|M|x\rangle$ in contradiction in HHL algorithm? How can we print the solution vector $|x\rangle$ without losing the exponential speedup?
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How to theoretically compare the complexity of quantum and classical algorithms?

I am working on reducing an NP class problem to a QUBO so can be solved with QAOA. I know that there is not a practical way to compare the performance as there is no QPU with enough qubits. I am doing ...
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Complexity of a distribution of measurement of output of quantum circuit

The Kolmogorov complexity of a string refers to a deterministic object. Here, I refer to the analogous "complexity of a distribution", or better, to the complexity of sampling from a ...
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Can a quantum computer tell whether a program is Turing complete?

I am very new to quantum computing and would like to know if a quantum computer can decide whether a given program is Turing complete.
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Quantum advantage with only Clifford gates (Gottesman Knill theorem)

Let's say I want to solve a computational task which input can be encoded in $n$ bits of information. The look for a quantum advantage is (usually) asking to find a quantum algorithm in which there ...
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2answers
333 views

What is the computational complexity in initializing a quantum register?

I'm trying to figure out what is the computational complexity of initializing a quantum register of N qubits. For my research, I have used the initialize method of qiskit, in which you set the ...
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Could quantum computers answer the question of whether QCD predicts quark gluon confinement?

As I understand it, it is not known whether or not QCD actually predicts quark gluon confinement. As I understand it answering questions in quantum field theories is generally harder in terms of ...
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68 views

What is quantum advantage truly?

Let's consider the Deutsch Jozsa algorithm, I understand that the superposition principle in quantum mechanics, helps us design circuits which would give answers in one single query. But then I would ...
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46 views

References about deriving the complexity of a given algorithm

Trying to learn about how to derive (& intuition) the complexity for a given algorithm as shown below. If there is any good reference or starting point that anyone can suggest that will be highly ...
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Is quantum query complexity equivalent to the total number of calls to the quantum computer for any given algorithm?

In other words, if an algorithm requires N total calls to the quantum computer to find the solution (of any given problem), would N be equivalent to its query complexity?
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How powerful are boundedly many $T$-gates?

For a natural number $k$ (0 is a natural number), let $T_k$ be the collection of all languages that can be efficiently decided by quantum circuits consisting of Clifford gates and at most $k$ $T$-...
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BHT algorithm implementation

Summary of Method Amplitude Amplification Summary The BHT algorithm uses amplitude amplification, a nice generalisation of Grover's algorithm, where there is a subset $G\subset X$ of good elements in ...
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Proving that with probability 1 $NP \nsubseteq BQP$ with respect to random oracles

In the paper Strength and Weakneses of Quantum Computers (https://arxiv.org/abs/quant-ph/9701001) by Bennet, Bernstein, Brassard and Vazirani, it is shown the statement in the title (Theorem 3.5 in ...
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How come classical Deutsch-Jozsa is $O(1)$ when allowing "a small error rate"?

I'm reading Quantum Computing: An Applied Approach, by Hidary. Chapter 8.2 (p104) says: While it is true that Deutsch-Jozsa demonstrates an advantage of quantum over classical computing, if we allow ...
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37 views

Comparing complexity of digital and analog quantum computation

The complexity of an algorithm run on a digital quantum computer is quantified, roughly, by the number of elementary gates in the corresponding circuit. Can one similarly quantify the complexity of an ...
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Is there an efficient quantum circuit that create a random permuntation matrix?

Suppose we want to generate a random, random according to some probability distribution, unitary permutation matrix that is applied to an input of $n$ qubits. So is there an efficient polynomial time ...
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Are all computational resources reducible to the time resource?

It's well known that in most (if not all?) computations you can trade time and space resources. An extreme example might be creating an infinitely large lookup table of all composites produced from ...
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35 views

Would quantum computers be more efficient at solving circular reference problems than classical computers?

A circular reference is when a certain value either refers to itself or a value refers to a value that refers to it. An example of a circular reference problem would be $x=f(x)$. One way to solve ...
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Postselection and hardness of estimating amplitudes

Let $A$ be a class of quantum circuits such that \begin{equation} \text{Post}A = \text{Post}BQP, \end{equation} where $\text{Post}$ indicates post-selection. Is only this amount of information ...
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2answers
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Why is depth complexity revelant?

Since gate complexity correspond to the number of gate for a given quantum circuit, it seems that depth complexity bring no more information about quantum complexity than gate complexity. So does gate ...
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Is there a practical architecture-independent benchmark suitable for adversarial proof of quantum supremacy?

Recent quantum supremacy claims rely, among other things, on extrapolation, which motivates the question in the title, where the word "adversarial" is added to exclude such extrapolation-...
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1answer
156 views

Spoofing XQUATH with the Feynman method

Consider the XQUATH conjecture for random quantum circuits, as mentioned here. (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time classical algorithm that ...
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Feynman method and polynomial time algorithm for XQUATH

Consider the Feynman algorithm for simulating quantum circuits, as given here. Consider the XQUATH conjecture for random quantum circuits from here, given by (XQUATH, or Linear Cross-Entropy Quantum ...
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Query complexity on Quantum Pattern Matching of Mateus Algorithm

I am trying to understand the complexity of the Mateus and Omar algorithm for quantum pattern matching, it is clear to me from the pseudocode that the query complexity is $O(\sqrt{N})$, apart from the ...
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Complexity of $n$-Toffoli with phase difference

I'm interested in the $n$-Toffoli gates with phase differences. I found a quadratic technique in section 7.2 of this paper. Here's the front page of the paper. Here's an image of the section that I'm ...
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Why is sampling considered difficult on a classical computer but easy on a quantum computer? [closed]

It is my understanding that classical computers have a hard time sampling results from an output from a quantum circuit, but quantum computers find it very easy to do so. Why is this?
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What are the types of models of computation aside from the quantum query model?

It looks like in a lot of quantum algorithms, we use the quantum query model. I wanted to know what are the other types of models of computation, used in quantum computing as well as in classical ...
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What constitutes generic dynamics, and how is it different from a fully random function?

What constitutes generic dynamics? And how is it different from a fully random function? From what I understand, a fully random function is one that is "Haar" random. And generic dynamics, ...
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1answer
49 views

Learning k positions of a Boolean function with a quantum computer

Consider a Boolean function with multiple outputs $f: \{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$, and consider being given oracle access to the function $f$. Let us denote the oracle by $O_f$. For an $x \...
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What is the counting argument for the number of elementary operations required for a random function?

What is the counting argument for the following statement (classical)? "A random function on n bits requires $e^{\Omega(n)}$ elementary operations." It appears in the introduction of PRL 116,...
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Accessing parameter in oracle and their relation

I am actually a newbie in quantum computing. I do have some doubts regarding quantum query complexity. From what I understood is that we can't explicitly give the input and use oracle for this purpose....
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Relation between approximate counting and sampling

Consider the following statement of Stockmeyer counting theorem. Given as input a function $f:\{0, 1\}^{n} \rightarrow \{0, 1\}^{m}$ and $y \in \{0, 1\}^{m}$, there is a procedure that runs in ...
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Can the difference between quantum and classical circuits be attributed to different paths in the Hilbert space?

One of the explanations I have encountered for why quantum computation can provide speed-up over the classical is a picture that in the Hilbert space much more paths are allowed quantum-mechanically ...
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When is a Quantum Computer Slower Than a Classical Computer?

Someone offhandedly mentioned to me that quantum computers are sometimes significantly (I guess they meant asymptotically) slower than classical computers. Unfortunately, I didn't get any arguments ...
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Is there a quantum implementation like HashSet?

There are many data structures in classical computers, like Tree, HashSet, etc. These data structures give convenience to the performance (time complexity) of algorithms. I am wondering how to create ...
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1answer
168 views

How can one cheat in Mahadev's classical verification protocol if one can find a "claw''?

I was going through the seminal paper of Urmila Mahadev on Classical Verification of Quantum Computations(for an overview see this excellent talk by her). As a physicist by training, I am not very ...
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What is the quantum query complexity of the period finding routine of Shor's algorithm?

It seems like it should be a function of N - O(log N), to minimise probability of getting a multiple of the period. However, Prof Preskill's lec notes mention: Thus we solve Period Finding if the ...
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Worst Case Asymptotic Complexity of Berstein-Vazirani

How would one determine the worst-case asymptotic complexity ($\theta$) of a Bernstein-Vazirani circuit encoding the secret 1111?
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Is the complexity of a quantum circuit constant in the depth of the circuit?

Take a quantum circuit on $n$ qubits, you have some sequence of gates. You can represent these gates as hermitian matrices, and then with some padding, you could take the product of these matrices, by ...
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How do I construct the oracle for a general Bernstein-Vazirani circuit?

Say I have a secret of length $n$, $s = |x_{n-1}x_{n-2}...x_0 \rangle$. If I want to construct an oracle for this problem would I just insert a CNOT gate on every qubit where the secret's value is 1? ...
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Estimating output amplitudes of quantum circuits as GapP functions

Let's fix a universal gate set comprising of a Hadamard gate and a Toffoli gate. Consider an $n$ qubit quantum circuit $U_{x}$, made up of gates from that universal set, applied to initial state $|0^{...
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Degree of $N$-bit Majority function is larger or equal to $N/2$

I am looking to prove that the $N$-bit Majority function $f$, which is 1 if its input $x \in \{0, 1\}^N$ has Hamming weight $> N/2$, and 0 if its input has Hamming weight $\leq N/2$ has degree $\...
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109 views

Quantum hardness of XQUATH conjecture

Consider the XQUATH conjectures, as defined here (https://arxiv.org/abs/1910.12085, Definition 1). (XQUATH, or Linear Cross-Entropy Quantum Threshold Assumption). There is no polynomial-time ...
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Are spin-glass problems NP (-complete)?

It is well known that finding ground states for spin-glass systems (Ising, XY...) is NP-hard (at least as hard as the hardest NP-problems) so that they can be efficiently used to solve other NP ...
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Complexity of Quantum Satisfiability vs Local Hamiltonians

$k$QSAT$c$ is the promise problem where the input, given in an explicit encoding with finite number of bits, is a set $\{p_{1},p_{2},\ldots p_{m}\}$ of $k$-local projectors over a $n$-qbits register, ...
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The complexity of LH restricted to projectors

Let's denote $kLP_{c}$ the promise problem where the input, given in an explicit encoding with finite number of bits, is a set $\{p_{1},p_{2},\ldots p_{m}\}$ of k-local projectors over a n-qbits ...