We introduce two decision problems, STABILIZERD and TRANSLATING COSETD, and give quantum reductions from them to the problem ORBIT SUPERPOSITION, as well as quantum reductions to them from two group theoretic problems GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP. Based on these reductions, efficient quantum algorithms are obtained for GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP in the setting of black-box groups. Specifically, for solvable groups, this gives efficient quantum algorithms for GROUP INTERSECTION if one of the underlying solvable groups has a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one of the underlying solvable groups is smoothly solvable. We also show that GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP are in the complexity class SZK.
We introduce two decision problems, STABILIZERD and TRANSLATING COSETD, and give quantum reductions from them to the problem ORBIT SUPERPOSITION, as well as quantum reductions to them from two group theoretic problems GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP. Based on these reductions, efficient quantum algorithms are obtained for GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP in the setting of black-box groups. Specifically, for solvable groups, this gives efficient quantum algorithms for GROUP INTERSECTION if one of the underlying solvable groups has a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one of the underlying solvable groups is smoothly solvable. We also show that GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP are in the complexity class SZK.
We study two group theoretic problems, GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, in the setting of black-box groups, where DOUBLE COSET MEMBERSHIP generalizes a set of problems, including GROUP MEMBERSHIP, GROUP FACTORIZATION, and COSET INTERSECTION. No polynomial-time classical algorithms are known for these problems. We show that for solvable groups, there exist efficient quantum algorithms for GROUP INTERSECTION if one of the underlying solvable groups has a smoothly solvable commutator subgroup, and for DOUBLE COSET MEMBERSHIP if one of the underlying solvable groups is smoothly solvable. We also study the decision versions of STABILIZER and ORBIT COSET, which generalizes GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP, respectively. We show that they reduce to ORBIT COSET under certain conditions. Finally, we show that DOUBLE COSET MEMBERSHIP and DOUBLE COSET NONMEMBERSHIP have zero knowledge proof systems.
We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups, including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Zn p, whenever p is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup and prove a powerful self-reducibility result: Translating Coset in a finite solvable group G is reducible to instances of Translating Coset in G/N and N, for appropriate normal subgroups N of G. Our self-reducibility framework, combined with Kuperberg’s subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.
In this paper, we present a polynomial-time quantum algorithm for computing a minimum-sized generating set of solvable black-box groups. Next, we consider the class Γd of black-box groups, where every non-abelian composition factor is isomorphic to a subgroup of the symmetric group Sd for a fixed d. We design polynomial-time quantum algorithms to compute the direct product decomposition of abelian factor groups and solve the constructive membership problem for factor groups of groups from Γd. With the help of these algorithms, we design a quantum algorithm for computing a chief series of black-box groups from Γd. Using the chief series, we construct a polynomial-time quantum algorithm for computing minimum generating sets of black-box groups from Γd. Finally, we show that the minimum generating set problem for general black-box groups is in NP∩coAM.
In the dissertation we study topics in quantum computation. First, we study the model of constant-depth quantum circuits. We study the family QNC0 of constant-depth quantum circuits, and the complexity classes associated with QNC0 circuits: EQNC0, NQNC0, and BQNC0e,d . We show certain containment results such as NQNC 0 = NQACC = NQP = coC =P and BQNC0e,d is in P for certain e and δ. Our results essentially refute a conjecture of Green et al. that NQACC ⊆ TC0. We also define and study complexity classes postEQP, postRQP, postNQP, and postBQNC0 We show containment results such as postBQNC0 = postBQP = PP and NQP = postRQP = postNQP.
Second, we study applications of quantum algorithms in computational group theory. We give results about quantum algorithms and reductions for group theoretic problems, concentrating mostly on solvable groups. We study two particular group theoretic problems—GROUP I NTERSECTION and DOUBLE COSET M EMBERSHIP. We show that these problems reduce to other group problems with known efficient quantum algorithms for many instances, yielding efficient quantum algorithms for GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP on the same types of groups. Then we generalize and refine our results by introducing decision versions of the STABILIZER and ORBIT COSET problems, and showing that these new problems lie in between GROUP INTERSECTION and DOUBLE COSET MEMBERSHIP on the one hand, and the problem ORBIT SUPERPOSITION, on the other. We also show that GROUP INTERSECTION and D OUBLE COSET MEMBERSHIP have statistical zero knowledge proofs. Finally we give an alternative quantum algorithm for the problem of decomposing finite abelian groups.
We study the computational complexity of quantum state isomorphism problems under group actions: given two quantum circuits that prepare pure or mixed states, decide whether the two states are related by a group action. This can be seen as a quantum state version of the Hidden Shift Problem, in much the same way that the State Hidden Subgroup Problem is a quantum version of the ordinary Hidden Subgroup Problem.
We prove several results for this computational problem:
- For the pure-state version, we show that the problem is BQP-hard for all nontrivial groups, and contained in QCMA ∩ QCSZK. We further obtain refined results for specific groups of interest: for abelian groups we show that the problem reduces to the state hidden subgroup problem over the generalized dihedral group; for the Clifford group, the problem is at least as hard as Graph Isomorphism under polynomial-time reductions; for the Pauli group it is BQP-complete.
- For the mixed-state version, for nontrivial, finite and efficiently representable groups, the problem is QSZK-complete.
- We also study a variant of this problem over an infinite group, in particular, the bosonic linear optical unitaries. We show that in the setting where the classical description of the quantum state is given in a suitable wave function representation known as the stellar representation, the problem is at least as hard as Graph Isomorphism, and is contained in NP ∩ SZK.
Prior to our work, state isomorphism problems had only been studied for the symmetric group [LG17]. As a consequence of our results, we resolve an open question posed in [HEC25] about the existence of a quantum algorithm for the abelian state hidden subgroup problem on mixed states. We show that this problem is QSZK-hard in the worst case, thereby ruling out an efficient quantum algorithm unless QSZK = BQP.
We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups, including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Zpn, whenever p is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup and prove a powerful self-reducibility result: Translating Coset in a finite solvable group G is reducible to instances of Translating Coset in G/N and N, for appropriate normal subgroups N of G. Our self-reducibility framework, combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.