Questions tagged [complexity-theory]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
13 views

Computational Complexity of random Transverse-Ising Chain

It is well known that many NP-hard classical problems can be mapped to a spin-configuration Ising problem (see for example https://arxiv.org/pdf/1302.5843.pdf) However, what I would like to know is ...
user avatar
8 votes
2 answers
188 views

A rigorous definition for an exponential quantum advantage

Let's assume that we have an algorithmic problem to solve. This problem takes an integer $n$ as input to describe it and provides as output a bit string providing the answer we are expecting. For some ...
user avatar
3 votes
1 answer
63 views

O(N log(M)) vs O(log(MN)) Complexity Name

I have a quantum system that solves a problem that takes $O(MN)$ on a classical computer. However, because it is solved using a quantum algorithm, it takes $O(\log(MN))$. I also have another algorithm ...
user avatar
1 vote
0 answers
74 views

What is the complexity of the Hadamard test and the SWAP test?

How to calculate the complexity of both the Hadamard test and SWAP test with $n$ qubits?
user avatar
0 votes
0 answers
58 views

How to estimate the complexity of a variational quantum circuit?

How to estimate the complexity of a variational quantum circuit? For example, I have a quantum circuit of $n$ qubits that uses alternating operators to build the network and train it. So what is the ...
user avatar
3 votes
1 answer
39 views

Do quantum algorithms and classical algorithms of the same complexity have the same time consumption?

The classical computer field likes to use the complexity O(x) to represent the complexity of an algorithm. Is this concept applicable to quantum computers, if quantum algorithm A with the same ...
user avatar
6 votes
2 answers
149 views

How can I show that $\mathsf{QMA}\subseteq \mathsf{PSPACE}$

Lately I have seen the claim that $\mathsf{QMA}\subseteq \mathsf{PSPACE}$, and I wonder how can it be proved. Thanks
user avatar
  • 119
6 votes
1 answer
261 views

What is the complexity of hidden subgroup problems?

It is often stated that some of the "hidden subgroup problems" can be efficiently solved by quantum computers if the group is abelian, while no efficient algorithm is known for the non-...
user avatar
0 votes
1 answer
83 views

To what class of complexity theory a problem for whom I can’t check the solution, ever, belong?

Let’s say I have a problem that can only be solved with a trapdoor, but regardless of whether the trapdoor is right or wrong, you can’t check if you have found the solution to the problem. To what ...
user avatar
3 votes
0 answers
76 views

Relation between geometric and discrete circuit complexity

Geometric complexity of a unitary, as introduced for example here https://arxiv.org/abs/quant-ph/0502070, measures the length of a geodesic connecting the identity matrix and a given unitary in the ...
user avatar
2 votes
0 answers
40 views

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, ...
user avatar
-1 votes
1 answer
37 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
user avatar
  • 1
6 votes
1 answer
135 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 ...
user avatar
  • 2,057
5 votes
1 answer
144 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? ...
user avatar
2 votes
0 answers
86 views

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?
user avatar
  • 31
4 votes
1 answer
91 views

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 ...
user avatar
  • 81
5 votes
1 answer
114 views

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 ...
user avatar
9 votes
1 answer
1k views

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.
user avatar
7 votes
3 answers
819 views

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 ...
user avatar
5 votes
2 answers
345 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 ...
user avatar
3 votes
0 answers
51 views

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 ...
user avatar
2 votes
1 answer
77 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 ...
user avatar
  • 203
1 vote
1 answer
48 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 ...
user avatar
2 votes
1 answer
31 views

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?
user avatar
5 votes
1 answer
131 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 $k$ $T$-...
user avatar
  • 207
5 votes
1 answer
179 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 ...
user avatar
4 votes
0 answers
42 views

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 ...
user avatar
  • 483
2 votes
2 answers
98 views

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 ...
user avatar
  • 3,771
2 votes
1 answer
48 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 ...
user avatar
2 votes
0 answers
108 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 ...
user avatar
  • 55
4 votes
2 answers
102 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 ...
user avatar
1 vote
1 answer
36 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 ...
user avatar
6 votes
0 answers
76 views

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 ...
user avatar
3 votes
2 answers
139 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 ...
user avatar
  • 55
5 votes
0 answers
43 views

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-...
user avatar
  • 306
4 votes
1 answer
159 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 ...
user avatar
3 votes
0 answers
48 views

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 ...
user avatar
3 votes
0 answers
46 views

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 ...
user avatar
7 votes
2 answers
283 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 ...
user avatar
  • 101
1 vote
0 answers
47 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?
user avatar
  • 87
1 vote
1 answer
75 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 ...
user avatar
  • 23
2 votes
0 answers
27 views

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, ...
user avatar
3 votes
1 answer
56 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 \...
user avatar
1 vote
1 answer
51 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,...
user avatar
1 vote
1 answer
65 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....
user avatar
  • 23
2 votes
0 answers
72 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 ...
user avatar
4 votes
2 answers
110 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 ...
user avatar
2 votes
0 answers
96 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 ...
user avatar
2 votes
1 answer
57 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 ...
user avatar
5 votes
1 answer
183 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 ...
user avatar
  • 221

1
2 3 4 5