arXiv subjects

Tejas Bhojraj

Publications and source records attributed to Tejas Bhojraj.

7 recordsLinked to original sources

arXiv2024

Von Neumann Entropy and Quantum Algorithmic Randomness

A state $ρ=(ρ_n)_{n=1}^{\infty}$ is a sequence such that $ρ_n$ is a density matrix on $n$ qubits. It formalizes the notion of an infinite sequence of qubits. The von Neumann entropy $H(d)$ of a density matrix $d$ is the Shannon entropy of its eigenvalue distribution. We show: (1) If $ρ$ is a computable quantum Schnorr random state then $\lim_n [H(ρ_n )/n] = 1$. (2) We define quantum s-tests for $s\in [0,1]$, show that $\liminf_n [H(ρ_n)/n]\geq \{ s: ρ$ is covered by a quantum s-test $\}$ for computable $ρ$ and construct states where this inequality is an equality. (3) If $\exists c \exists^\infty n H(ρ_n)> n-c$ then $ρ$ is strong quantum random. Strong quantum randomness is a randomness notion which implies quantum Schnorr randomness relativized to any oracle. (4) A computable state $(ρ_n)_{n=1}^{\infty}$ is quantum Schnorr random iff the family of distributions of the $ρ_n$'s is uniformly integrable. We show that the implications in (1) and (3) are strict.

quant-ph↗

arXiv2021

Algorithmic Randomness and Kolmogorov Complexity for Qubits

Nies and Scholz defined quantum Martin-Löf randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum Schnorr randomness is then introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states. We introduce quantum-K, ($QK$) a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of $QK$. Several connections between Solovay randomness and $K$ carry over to those between weak Solovay randomness and $QK$. We then define $QK_C$, using computable measure machines and connect it to quantum Schnorr randomness. We then explore a notion of `measuring' a state. We formalize how `measurement' of a state induces a probability measure on the space of infinite bitstrings. A state is `measurement random' ($mR$) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-Löf randoms. I.e., measuring a $mR$ state produces a Martin-Löf random bitstring almost surely. While quantum-Martin-Löf random states are $mR$, the converse fails: there is a $mR$ state, $ρ$ which is not quantum-Martin-Löf random. In fact, something stronger is true. While $ρ$ is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. So, classical randomness can be generated from a computable state which is not quantum random. We conclude by studying the asymptotic von Neumann entropy of computable states.

quant-ph↗

arXiv2021

Notions of indifference for genericity: Union and subsequence sets

A set $I$ is said to be a universal indifferent set for $1$-genericity if for every $1$-generic $G$ and for all $X \subseteq I$, $G ΔX$ is also $1$-generic. Miller showed that there is no infinite universal indifferent set for $1$-genericity. We introduce two variants (union and subsequence sets for $1$-genericity) of the notion of universal indifference and prove that there are no non-trivial universal sets for $1$-genericity with respect to these notions. In contrast, we show that there is a non-computable subsequence set for weak-$1$-genericity.

math.LO↗

arXiv2021

Prefix-free quantum Kolmogorov complexity

We introduce quantum-K ($QK$), a measure of the descriptive complexity of density matrices using classical prefix-free Turing machines and show that the initial segments of weak Solovay random and quantum Schnorr random states are incompressible in the sense of $QK$. Many properties enjoyed by prefix-free Kolmogorov complexity ($K$) have analogous versions for $QK$; notably a counting condition. Several connections between Solovay randomness and $K$, including the Chaitin type characterization of Solovay randomness, carry over to those between weak Solovay randomness and $QK$. We work towards a Levin-Schnorr type characterization of weak Solovay randomness in terms of $QK$. Schnorr randomness has a Levin-Schnorr characterization using $K_C$; a version of $K$ using a computable measure machine, $C$. We similarly define $QK_C$, a version of $QK$. Quantum Schnorr randomness is shown to have a Levin-Schnorr and a Chaitin type characterization using $QK_C$. The latter implies a Chaitin type characterization of classical Schnorr randomness using $K_C$.

quant-ph↗

arXiv2021

Quantum algorithmic randomness

Quantum Martin-Löf randomness (q-MLR) for infinite qubit sequences was introduced by Nies and Scholz. We define a notion of quantum Solovay randomness which is equivalent to q-MLR. The proof of this goes through a purely linear algebraic result about approximating density matrices by subspaces. We then show that random states form a convex set. Martin-Löf absolute continuity is shown to be a special case of q-MLR. Quantum Schnorr randomness is introduced. A quantum analogue of the law of large numbers is shown to hold for quantum Schnorr random states.

quant-ph↗

arXiv2020

Global Optimum Search in Quantum Deep Learning

This paper aims to solve machine learning optimization problem by using quantum circuit. Two approaches, namely the average approach and the Partial Swap Test Cut-off method (PSTC) was proposed to search for the global minimum/maximum of two different objective functions. The current cost is $O(\sqrt{|Θ|} N)$, but there is potential to improve PSTC further to $O(\sqrt{|Θ|} \cdot sublinear \ N)$ by enhancing the checking process.

quant-ph↗

arXiv2020

Generating Randomness from a Computable, Non-random Sequence of Qubits

Nies and Scholz introduced the notion of a state to describe an infinite sequence of qubits and defined quantum-Martin-Lof randomness for states, analogously to the well known concept of Martin-Löf randomness for elements of Cantor space (the space of infinite sequences of bits). We formalize how 'measurement' of a state in a basis induces a probability measure on Cantor space. A state is 'measurement random' (mR) if the measure induced by it, under any computable basis, assigns probability one to the set of Martin-Löf randoms. Equivalently, a state is mR if and only if measuring it in any computable basis yields a Martin-Löf random with probability one. While quantum-Martin-Löf random states are mR, the converse fails: there is a mR state, x which is not quantum-Martin-Löf random. In fact, something stronger is true. While x is computable and can be easily constructed, measuring it in any computable basis yields an arithmetically random sequence with probability one. I.e., classical arithmetic randomness can be generated from a computable, non-quantum random sequence of qubits.

cs.IT↗