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Hardness of Efficiently Generating Ground States in Postselected PHYSICAL REVIEW RESEARCH 3, 013213 (2021) Hardness of efficiently generating ground states in postselected quantum computation Yuki Takeuchi ,* Yasuhiro Takahashi, and Seiichiro Tani NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan (Received 30 June 2020; accepted 23 February 2021; published 8 March 2021) Generating ground states of any local Hamiltonians seems to be impossible in quantum polynomial time. In this paper, we give evidence for the impossibility by applying an argument used in the quantum-computational- supremacy approach. More precisely, we show that if ground states of any 3-local Hamiltonians can be approximately generated in quantum polynomial time with postselection, then PP = PSPACE. Our result is superior to the existing findings in the sense that we reduce the impossibility to an unlikely relation between classical complexity classes. We also discuss what makes efficiently generating the ground states hard for postselected quantum computation. DOI: 10.1103/PhysRevResearch.3.013213 I. INTRODUCTION longer time than quantum complexity classes, unlikely re- lations between classical complexity classes would be more Quantum computing is expected to outperform classical dramatic than those involving quantum complexity classes. computing. Indeed, quantum advantages have already been As subuniversal quantum computing models showing quan- shown in terms of query complexity [1] and communica- tum computational supremacy, several models have been tion complexity [2]. Regarding time complexity, it is also proposed, such as boson sampling [5–7], instantaneous quan- believed that universal quantum computing has advantages tum polynomial time (IQP) [8,9], and its variants [10–13], over classical counterparts. For example, although an efficient deterministic quantum computation with one quantum bit quantum algorithm, i.e., Shor’s algorithm, exists for integer (DQC1) [14,15], Hadamard-classical circuit with one qubit factorization [3], there is no known classical algorithm that (HC1Q) [16], and quantum random circuit sampling [17–20]. can do so efficiently. However, an unconditional proof that A proof-of-principle demonstration of quantum computa- there is no such classical algorithm seems to be hard because tional supremacy has recently been achieved using quantum an unconditional separation between BQP and BPP implies random circuit sampling with 53 qubits [21]. Regarding other P = PSPACE. Whether P = PSPACE is a long-standing models, small-scale experiments have been performed toward problem in the field of computer science. the goal of demonstrating quantum computational supremacy To give evidence of quantum advantage in terms of compu- [22–27]. tational time, a sampling approach has been actively studied. On the other hand, the limitations of universal quantum This approach is to show that if the output probability dis- computing are also actively studied (e.g., see Refs. [28–30]). tributions from a family of (nonuniversal) quantum circuits Understanding these limitations is important to clarify how to can be efficiently simulated in classical polynomial time, make good use of universal quantum computers. For example, then the polynomial hierarchy (PH) collapses to its second it is believed to be impossible in the worst case to gener- or third level. Since it is widely believed that PH does not ate ground states of a given local Hamiltonian in quantum collapse, this approach shows one kind of quantum advantage polynomial time, while their heuristic generation has been (under a plausible complexity-theoretic assumption). This studied using quantum annealing [31], variational quantum type of quantum advantage is called quantum computational eigensolvers (VQE) [32], and quantum approximate optimiza- supremacy [4]. The quantum-computational-supremacy ap- tion algorithms (QAOA) [33]. Since deciding whether the proach is remarkable because it reduces the impossibility of ground-state energy of a given 2-local Hamiltonian is low or an efficient classical simulation of quantum computing to high with polynomial precision is a QMA-complete problem unlikely relations between classical complexity classes (under [34], if efficient generation of the ground states is possible, conjectures such as the average-case hardness conjecture). then BQP = QMA that seems to be unlikely. As well as Since classical complexity classes have been studied for a the gap between quantum and classical computing in terms of time complexity, it is hard to unconditionally show the impossibility of efficiently generating the ground states. *[email protected] In this paper, we utilize a technique from the quantum- computational-supremacy approach to give new evidence for Published by the American Physical Society under the terms of the this impossibility. More precisely, in Theorem 1,weshow Creative Commons Attribution 4.0 International license. Further that if the ground states of any given 3-local Hamiltoni- distribution of this work must maintain attribution to the author(s) ans can be approximately generated in quantum polynomial and the published article’s title, journal citation, and DOI. time with postselection, then PP = PSPACE. Similar to the 2643-1564/2021/3(1)/013213(8) 013213-1 Published by the American Physical Society TAKEUCHI, TAKAHASHI, AND TANI PHYSICAL REVIEW RESEARCH 3, 013213 (2021) quantum-computational-supremacy approach, this conse- (a) o quence leads to the collapse of a hierarchy, i.e., the counting = p hierarchy (CH) collapses to its first level (CH PP). In |0m { } Theorem 2, we consider a different notion of approximation x pz z∈{0,1}nm +m ... and show that if the probability distributions obtained from U ... the ground states can be approximately generated in quantum ⊗m nm |g polynomial time with postselection, then PP = PSPACE. (b) Theorem 2 studies the hardness of approximately generating o the ground states from a different perspective, because it is p closely related to the hardness of approximately generating |0m the probability distributions. Furthermore, by using a similar x ... U ... argument, we show that if the ground states of any 3-local ⊗m nm Hamiltonians can be specified by using polynomial numbers ρapprox of bits, then NPPP = PSPACE. This leads to the second-level PP m ⊗m collapse of CH, i.e., CH = PP . This result seems to give FIG. 1. (a) A quantum circuit Ux with an input state |0 |g to additional evidence to support the conclusion that QMA is decide whether x ∈ L or x ∈/ L,whereL is in postQMA.Leto and strictly larger than QCMA. Here we say that a ground state is p be output and postselection registers, respectively. If o = p = 1, specified by using a polynomial number of bits if there exists a we conclude that x ∈ L. On the other hand, if p = 1ando = 0, polynomial-size quantum circuit that outputs the ground state then x ∈/ L. The output probability distribution of nm + m qubits is { } given the success of the postselection. Since ordinary univer- denoted by pz z∈{0,1}nm +m . Each meter symbol represents a Z-basis sal gate sets contain only a constant number of elementary measurement. (b) The same quantum circuit as in (a) except that |g is replaced with an approximate state ρapprox. The output and gates, we can specify any polynomial-size quantum circuit by using a polynomial number of bits. Our results are different postselection registers are denoted by o and p , respectively. from the existing ones on the impossibility of efficient ground- state generation in a sense that we reduce the impossibility to unlikely relations between classical complexity classes as in in several papers such as Ref. [37]. At least, we can say that it the quantum-computational-supremacy approach. is difficult to show that PH ⊆ PP holds. This is because there This paper is organized as follows. In Sec. II,wegivesome exists an oracle relative to which PH (more precisely, PNP)is preliminaries on complexity classes and show an important not contained in PP [38]. lemma (Lemma 1) that is used to obtain our theorems. In The complexity class postQMA is defined as follows Sec. III, as the first main result, we show that it is hard for [39,40]: a language L is in postQMA if and only if there postselected quantum computers to approximately generate exist a constant 0 <δ<1/2, polynomials n, m, and k, and ground states of a given 3-local Hamiltonian in the worst a uniform family {Ux}x of polynomial-size quantum circuits, case (Theorem 1). In Sec. IV, as the second main result, we where x is an instance, and Ux takes an n-qubit state ρ and an- − show that it is also hard for postselected quantum computers cillary qubits |0m as inputs, such that (i) Pr[p = 1 | ρ] 2 k, to approximately generate probability distributions obtained where p is a single-qubit postselection register, for any ρ, (ii) from the ground states, under a different notion of approxi- if x ∈ L, then there exists a witness ρx such that Pr[o = 1 | p = mation (Theorem 2). Section V is devoted to conclusion and 1,ρx] 1/2 + δ with a single-qubit output register o, and (iii) discussion. if x ∈/ L, then for any ρ,Pr[o = 1 | p = 1,ρ] 1/2 − δ.In this definition, “polynomials” mean the ones in the length | | II. PRELIMINARIES x of the instance x. Note that postQMA is denoted by QMApostBQP in Ref. [39]. Before we explain our results, we will briefly review The following is an important lemma: preliminaries required to understand our argument. We use Lemma 1. Any decision problem in postQMA can be ef- several complexity classes that are sets of decision problems. ficiently solved using postselected polynomial-size quantum Here decision problems are mathematical problems that can circuits if a polynomial number of copies of a ground state be answered by YES or NO. We mainly use complexity (i.e., a minimum-eigenvalue state) |g of an appropriate 3- classes CH, postBQP, postQCMA, and postQMA, where local Hamiltonian is given [see Fig.
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