Efficiently generating ground states is hard for postselected quantum computation Yuki Takeuchi,1, ∗ Yasuhiro Takahashi,1 and Seiichiro Tani1 1NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan Although quantum computing is expected to outperform universal classical computing, an unconditional proof of this assertion seems to be hard because an unconditional separation between BQP and BPP implies P 6= PSPACE. Because of this, the quantum-computational-supremacy approach has been actively studied; it shows that if the output probability distributions from a family of quantum circuits can be efficiently simulated in classical polynomial time, then the polynomial hierarchy collapses to its second or third level. Since it is widely believed that the polynomial hierarchy does not collapse, this approach shows one kind of quantum advantage under a plausible assumption. On the other hand, the limitations of universal quantum computing are also actively studied. For example, it is believed to be impossible to generate ground states of any local Hamiltonians in quantum polynomial time. In this paper, we give evidence for this 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 the counting hierarchy collapses to its first level. Our evidence is superior to the existing findings in the sense that we reduce the impossibility to an unlikely relation between classical complexity classes. Furthermore, our argument can be used to give evidence that at least one 3-local Hamiltonian exists such that its ground state cannot be represented by a polynomial number of bits, which may be related to a gap between QMA and QCMA. Quantum computing is expected to outperform classical 13], deterministic quantum computation with one quantum bit computing. Indeed, quantum advantages have already been (DQC1) [14, 15], Hadamard-classical circuit with one qubit shown in terms of query complexity [1] and communication (HC1Q) [16], and quantum random circuit sampling [17– complexity [2]. Regarding time complexity, it is also be- 20]. A proof-of-principle demonstration of quantum compu- lieved that universal quantum computing has advantages over tational supremacy has recently been achieved using quan- classical counterparts. For example, although an efficient tum random circuit sampling with 53 qubits [21]. Regard- quantum algorithm, i.e., Shor’s algorithm, exists for integer ing other models, small-scale experiments have been per- factorization [3], there is no known classical algorithm that formed toward the goal of demonstrating quantum computa- can do so efficiently. However, an unconditional proof that tional supremacy [22–27]. there is no such classical algorithm seems to be hard because On the other hand, the limitations of universal quantum an unconditional separation between BQP and BPP implies computing are also actively studied (e.g., see Refs. [28–30]). P PSPACE. Whether P PSPACE is a long-standing = = Understanding these limitations is important to clarify how to problem6 in the field of computer6 science. make good use of universal quantum computers. For example, it is believed to be impossible to generate ground states of any To give evidence of quantum advantage in terms of com- local Hamiltonians in quantum polynomial time, while their putational time, a sampling approach has been actively stud- heuristic generation has been studied using quantum anneal- ied. This approach is to show that if the output probabil- ing [31], variational quantum eigensolvers (VQE) [32], and ity distributions from a family of (non-universal) quantum quantum approximate optimization algorithms (QAOA) [33]. circuits can be efficiently simulated in classical polynomial Since deciding whether the ground-state energy of a given - time, then the polynomial hierarchy (PH) collapses to its 2 local Hamiltonian is low or high with polynomial precision is second or third level. Since it is widely believed that PH a QMA-complete problem [34], if efficient generation of the arXiv:2006.12125v1 [quant-ph] 22 Jun 2020 does not collapse, this approach shows one kind of quantum ground states is possible, then BQP QMA that seems to advantage (under a plausible complexity-theoretic assump- = be unlikely. This unlikeliness can be strengthened using the tion). This type of quantum advantage is called quantum complexity class PQMA[log] that includes QMA [35]. Simu- computational supremacy [4]. The quantum-computational- lating a single-qubit measurement on ground states of -local supremacy approach is remarkable because it reduces the 5 Hamiltonians is PQMA[log]-complete [36]. Therefore, if effi- impossibility of an efficient classical simulation of quantum cient generation is possible, then BQP PQMA[log] that would computing to unlikely relations between classical complexity = be more unlikely than BQP QMA. As well as the gap be- classes. Since classical complexity classes have been stud- = tween quantum and classical computing in terms of time com- ied for a longer time than quantum complexity classes, un- plexity, it is hard to unconditionally show the impossibility of likely relations between classical complexity classes would efficiently generating the ground states. be more unlikely than those involving quantum complexity classes. As sub-universal quantum computing models show- In this paper, we utilize a technique from the quantum- ing quantum computational supremacy, several models have computational-supremacy approach to give new evidence for been proposed, such as boson sampling [5–7], instantaneous this impossibility. More precisely, in Theorem 1, we show quantum polynomial time (IQP) [8, 9] and its variants [10– that if the ground states of any 3-local Hamiltonians can be 2 approximately generated in quantum polynomial time with (a) o postselection, then the counting hierarchy (CH) collapses to p m its first level, i.e., CH = PP. In Theorem 2, we consider a dif- |0 ! {p } nm ′+m ferent notion of approximationand show that if the probability Ux z z∈{0,1} . distributions obtained from the ground states can be approx- ′ m′ nm imately generated in quantum polynomial time with postse- |g!⊗ lection, then CH = PP. Theorem 2 studies the hardness of (b) ′ approximately generating the ground states from a different o p′ perspective, because it is closely related to the hardness of ap- m |0 ! proximately generating the probability distributions. Further- Ux . more, by using a similar argument, we show that if the ground . ′ states of any 3-local Hamiltonians can be uniquely repre- ⊗m′ nm sented by a polynomial number of bits, then CH collapses ρapprox PP to its second level, i.e., CH = PP . This result seems to ′ m ⊗m give additional evidence to support the conclusion that QMA FIG. 1: (a) A quantum circuit Ux with an input state |0 i|gi to is strictly larger than QCMA. Our results are different from decide whether x ∈ L or x∈ / L, where L is in postQMA. Let o and be output and postselection registers, respectively. If , the existing ones on the impossibility of efficient ground-state p o = p = 1 we conclude that x ∈ L. On the other hand, if p = 1 and o = 0, generation in a sense that we reduce the impossibility to un- then x∈ / L. The output probability distribution of nm′ + m qubits likely relations between classical complexity classes as in the ′ is denoted by {pz}z∈{0,1}nm +m . Each meter symbol represents a quantum-computational-supremacy approach. Z-basis measurement. (b) The same quantum circuit as in Fig. 1 (a) Preliminaries.—Before we explain our results, we will except that |gi is replaced with an approximate state ρapprox. The briefly review preliminaries required to understand our ar- output and postselection registers are denoted by o′ and p′, respec- gument. We use several complexity classes that are sets tively. of decision problems. Here, decision problems are mathe- matical problems that can be answered by YES or NO. We circuits if a polynomial number of copies of a ground state mainly use complexity classes CH, postBQP, postQCMA, (i.e., a minimum-eigenvalue state) of an appropriate - and postQMA, where the latter three are postselected versions g 3 local Hamiltonian is given (see Fig. 1| (a)).i Note that a -local of BQP, QCMA, and QMA, respectively. We assume that 3 Hamiltonian H = t H(i) with a polynomial t is the sum readers know the major complexity classes, such as P, PP, i=1 of polynomially many Hermitian operators (i) t , each PSPACE, and PH (for their definitions, see Ref. [37]). H i=1 of which acts on atP most three (possibly geometrically{ } nonlo- The class CH is the union of classes C P over all non- k cal) qubits. The operator norm H(i) is upper-bounded by negative integers k, i.e., CH = k≥0CkP, where C0P = P || || ∪ one for any 1 i t. and C P = PPCk P for all k 0. We say that CH col- ≤ ≤ k+1 ≥ lapses to its k-th level when CH = CkP. The first-level col- This lemma can be obtained by combiningresults in Refs. [40, lapse of CH seems to be especially unlikely. This is because, 42]. The proof is given in Appendix A. PP from Toda’s theorem [38], PH P CH. Therefore, if By removing and from the definition of postQMA, ⊆ ⊆ ρx ρ CH = PP, then PH PP. Since there exists an oracle rel- the complexity class postBQP is defined. Since PP = ⊆ NP ative to which PH (more precisely, P ) is not contained in postBQP [43], readers can replace PP with postBQP if they PP [39], the inclusion PH PP seems to be unlikely. are not familiar with the definition of PP. Furthermore, the ⊆ The complexity class postQMA is defined as follows [40, class postQCMA is defined by replacing each quantum state 41]: a language L is in postQMA if and only if there exist ρx and ρ with a polynomial number of classical bits.