Questions tagged [complexity-theory]

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

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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
315 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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1answer
57 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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1answer
42 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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1answer
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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?
5
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1answer
113 views

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 $k$ $T$-...
5
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1answer
117 views

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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2answers
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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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1answer
35 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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0answers
88 views

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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2answers
85 views

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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1answer
32 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
74 views

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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0answers
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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
154 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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2answers
226 views

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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39 views

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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1answer
63 views

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
48 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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1answer
44 views

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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1answer
57 views

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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47 views

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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2answers
101 views

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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84 views

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 ...
2
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1answer
51 views

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
147 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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1answer
94 views

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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1answer
46 views

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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1answer
113 views

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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1answer
101 views

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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2answers
114 views

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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1answer
46 views

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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1answer
102 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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3answers
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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 ...
2
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1answer
98 views

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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1answer
63 views

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 ...
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1answer
148 views

How is a promise gap related to a spectral gap?

In linear algebra one often concerns oneself with the spectral gap of a given matrix, which may be defined as the difference between the smallest and second-smallest eigenvalue (or, depending on ...
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2answers
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The complexity of LH with constant gap

Kitaev's quantum equivalent of the Cook-Levin Theorem, provides a polynomial time classical reduction from a QMA verification circuit to a sum $H$ of local hamiltonians, such that the least eigenvalue ...
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1answer
104 views

How would I theorise a quantum query algorithm in O(1)?

I am currently attempting to solve a problem from Nielsen-Chuang, and I can't seem to figure out how I would do this; I'm trying to implement Grover's algorithm to solve the problem of differentiating ...
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1answer
1k views

Can quantum computer solve NP-complete problems? [duplicate]

As far as I know, quantum computers are able to solve only some of the NP-Problems in polynomial time, using the Grovers algorithm. I read that if one manages to create a reduction of Grovers ...
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1answer
357 views

Classical algorithm with complexity similar to Shor's discovered: Are there more efficient quantum algorithms than Shor's?

In the article Fast Factoring Integers by SVP Algorithms the author claims that he discovered classical algorithm for factoring integers in polynomial time. The Quantum Report mentioned here that it ...
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1answer
59 views

What does "large-scale universal quantum computation" mean?

I was reading this lecture notes by Sevag Gharibian. He mentioned the following statement: Large-scale universal quantum computer can be built. I do understand what quantum computer is, but I don't ...
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1answer
90 views

How do I calculate the number of uses of a unitary $U$ in iterative phase estimation?

How would one go along to calculate the number of uses of an unitary $U$ in Iterative Phase Estimation (IPE) to compare it to the number of uses of $U$ in standard Phase Estimation (Qiskit QPE)?
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Why does QAOA achieve quantum supremacy in an algorithmic sense?

In the paper Quantum Supremacy through the Quantum Approximate Optimization Algorithm the authors claim (last sentence of page 15): "If [...] the QAOA outperforms all known classical algorithms ...