We study the problem of determining if the mode of the output distribution of a quantum circuit given as a black-box is larger than a given threshold. We design a quantum algorithm for a promised version of this problem whose space complexity is logarithmic in the size of the domain of the distribution. Developing on top of that we further design an algorithm to estimate the largest probability among the outcomes of that circuit. This allows to revisit a few recently studied problems in the few-qubits scenario, namely $k$-distinctness and its gapped version, estimating the largest frequency in an array, and estimating the min-entropy of a distribution. In particular, our algorithm for $k$-distinctness on $n$-sized $m$-valued arrays requires $O(\log n + \log m)$ qubits compared to $\Omega(poly(n))$ qubits required by all the previous algorithms, and its query complexity is optimal for $k=\Omega(n)$. We also study reductions between the above problems and derive better lower bounds for some of them. The time-complexities of our algorithms have a small overhead over their query complexities making them efficiently implementable on currently available quantum backends.
We study the problem of determining if the mode of the output distribution of a quantum circuit given as a black-box is larger than a given threshold. We design a quantum algorithm for a promised version of this problem whose space complexity is logarithmic in the size of the domain of the distribution. Developing on top of that we further design an algorithm to estimate the largest probability among the outcomes of that circuit. This allows to revisit a few recently studied problems in the few-qubits scenario, namely k-distinctness and its gapped version, estimating the largest frequency in an array, and estimating the min-entropy of a distribution. In particular, our algorithm for k-distinctness on n-sized m-valued arrays requires O(logn+logm) qubits compared to Ω(poly(n)) qubits required by all the previous algorithms, and its query complexity is optimal for k=Ω(n). We also study reductions between the above problems and derive better lower bounds for some of them. The time-complexities of our algorithms have a small overhead over their query complexities making them efficiently implementable on currently available quantum backends.
We introduce the problem of determining if the mode of the output distribution of a quantum circuit (given as a black-box) is larger than a given threshold, named HighDist, and a similar problem based on the absolute values of the amplitudes, named HighAmp. We design quantum algorithms for promised versions of these problems whose space complexities are logarithmic in the size of the domain of the distribution, but the query complexities are independent. Using these, we further design algorithms to estimate the largest probability and the largest amplitude among the output distribution of a quantum black-box. All of these allow us to improve the query complexity of a few recently studied problems, namely, k-distinctness and its gapped version, estimating the largest frequency in an array, estimating the min-entropy of a distribution, and the non-linearity of a Boolean function, in the O~(1)-qubits scenario. The time-complexities of almost all of our algorithms have a small overhead over their query complexities making them efficiently implementable on currently available quantum backends.
Amplitude filtering is concerned with identifying basis-states in a superposition whose amplitudes are greater than a specified threshold; probability filtering is defined analogously for probabilities. Given the scarcity of qubits, the focus of this work is to design log-space algorithms for them. Both algorithms follow a similar pattern of estimating the amplitude (or, probability for the latter problem) of each state, in superposition, then comparing each estimate against the threshold for setting up a flag qubit upon success, finally followed by amplitude amplification of states in which the flag is set. We show how to implement each step using very few qubits by designing three subroutines. Our first algorithm performs amplitude amplification even when the "good state'' operator has a small probability of being incorrect -- here we improve upon the space complexity of the previously known algorithms. Our second algorithm performs "true amplitude estimation'' in roughly the same complexity as that of "amplitude estimation'', which actually estimates a probability instead of an amplitude. Our third algorithm is for performing amplitude estimation in parallel (superposition) which is difficult when each estimation branch involves different oracles. As an immediate reward, we observed that the above algorithms for the filtering problems directly improved the upper bounds on the space-bounded query complexity of problems such as non-linearity estimation of Boolean functions and k-distinctness.
Estimating statistical properties of probability distributions is a fundamental problem in statistics and theoretical computer science. We develop quantum algorithms for estimating the q-Tsallis entropy HqTsa(P)=1−q∑i=1npiq−1 when 0<q<1, given purified quantum query access to a distribution P on [n]. When 0<q≤21, we provide a quantum algorithm with query complexity $\widetilde{O}\rbra{\left(\frac{n}{\varepsilon}\right)^{\frac{1}{2q}}}$, which is optimal up to polylogarithmic factors. When 21<q<1, we introduce a two-stage simultaneous probability estimation procedure, yielding a quantum algorithm with query complexity O(ε2q1n21+2q1+εn1−q), which improves the result given by Chen et al. (ICALP 2026) when $\varepsilon=O\rbra{n^{1-2q}}$.
A leading proposal for verifying near-term quantum supremacy experiments on noisy random quantum circuits is linear cross-entropy benchmarking. For a quantum circuit C on n qubits and a sample z∈{0,1}n , the benchmark involves computing |⟨z|C|0n⟩|2 , i.e. the probability of measuring z from the output distribution of C on the all zeros input. Under a strong conjecture about the classical hardness of estimating output probabilities of quantum circuits, no polynomial-time classical algorithm given C can output a string z such that |⟨z|C|0n⟩|2 is substantially larger than 12n (Aaronson and Gunn, 2019). On the other hand, for a random quantum circuit C , sampling z from the output distribution of C achieves |⟨z|C|0n⟩|2≈22n on average (Arute et al., 2019).In analogy with the Tsirelson inequality from quantum nonlocal correlations, we ask: can a polynomial-time quantum algorithm do substantially better than 22n ? We study this question in the query (or black box) model, where the quantum algorithm is given oracle access to C . We show that, for any ε≥1poly(n) , outputting a sample z such that |⟨z|C|0n⟩|2≥2+ε2n on average requires at least Ω(2n/4poly(n)) queries to C , but not more than O(2n/3) queries to C , if C is either a Haar-random n -qubit unitary, or a canonical state preparation oracle for a Haar-random n -qubit state. We also show that when C samples from the Fourier distribution of a random Boolean function, the naive algorithm that samples from C is the optimal 1-query algorithm for maximizing |⟨z|C|0n⟩|2 on average.
How much memory is needed to retain the quantum speedup for collision finding? For a uniformly random function f:[N]→[N], the BHT algorithm finds a collision using O(N1/3) queries and a quantumly accessible classical table containing O(N1/3) input-output pairs, whereas a logarithmic-space Grover search uses O(N) queries. Determining the optimal query-space tradeoff between these extremes remains a major open problem.
We resolve this equation within the class of label-symmetric algorithms, which treat the function f's output labels as interchangeable. We prove that such algorithm that makes T queries, uses S qubits, and finds a collision in a uniformly random function f:[M]→[N] with constant probability satisfies T=Ω(N1/3)andT2S=Ω(NlogN). For the setting where M=N, these bounds are matched by a space-efficient implementation of the BHT algorithm. As a consequence of our tradeoff, any label-symmetric algorithm for the search version of Element Distinctness on f:[n]→[n2] must satisfy T=Ω(n2/3)andT2S=Ω(n2logn), matching Ambainis's quantum walk. Thus, both tradeoffs are optimal within the class of label-symmetric algorithms.
To prove these results, we develop a space-sensitive version of the compressed oracle technique. The compressed oracle records the information learned by the algorithm in an evolving superposition of databases. Using label symmetry and representation theory, we show that an algorithm using S qubits can effectively retain information about only O(S/logN) collision-free database entries. Substituting this estimate into the compressed oracle technique yields the stated tradeoffs.
Quantum computing has attracted a lot of attention in different research fields, such as mathematics, physics and computer science. Quantum algorithms can solve certain problems significantly faster than classical algorithms. There are many numerical problems, especially those arising from quantum systems, which are notoriously difficult to solve using classical computers, since the computational time required often scales exponentially with the size of the problem. However, quantum computers have the potential to solve these problems efficiently, which is also one of the founding ideas of the field of quantum computing. In this thesis, we explore five computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm based on the Berry phases for the counting problem. For the running time, it is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result should be robust to decoherence and resilient to certain kinds of noise. Since the counting problem is the foundation of many other numerical problems, such as high-dimensional integration and path integration, our adiabatic algorithms can be directly generalized to these kinds of problems. In addition, we study the quantum PAC learning model, offering an improved lower bound on the query complexity. The lower bound is very close to the best lower bound on query complexity known for the classical PAC learning model. We also study the algorithms and the cost of simulating a system evolving under a given Hamiltonian. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required. Moreover, we derive the optimal order of convergence given the required error bound. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for Hamiltonian simulation. The evolution is simulated by a product of exponentials in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. We provide a scheme to bound the error of the final quantum state in a randomized algorithm, and obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. We also apply Hamiltonian simulation in estimating the ground state energy of a multiparticle system, which is also known as the multivariate Sturm-Liouville eigenvalue problem. Since the cost of this problem grows exponentially with the number of particles using deterministic classical algorithms, it suffers from the curse of dimensionality. Quantum computers can vanquish the curse, and we exhibit a quantum algorithm whose total cost are linear in the number of particles.