Pinned QMA: The power of fixing a few in proofs

Daniel Nagaj,1, ∗ Dominik Hangleiter,2 Jens Eisert,2, 3 and Martin Schwarz2 1RCQI, Institute of Physics, Slovak Academy of Sciences, Bratislava, Slovakia 2Dahlem Center for Complex Quantum Systems, Physics Department, Freie Universität Berlin, Germany 3Department of Mathematics and Computer Science, Freie Universität Berlin, Germany What could happen if we pinned a single of a system and fixed it in a particular state? First, we show that this leads to difficult static questions about ground state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the Pinned Commuting and Pinned Stoquastic Local Hamiltonian problems are QMA-. Second, we investigate pinned dynamics and demonstrate that fixing a single qubit via often repeated measurements results in universal quantum computation with commuting Hamiltoni- ans. Finally, we discuss variants of the Ground State Connectivity problem in light of pinning, and show that Stoquastic GSCON is QCMA-complete.

I. INTRODUCTION small subsystem. Orsucci et al. [41] have formulated the - lated question of Hamiltonian purification, investigating uni- The goal of quantum Hamiltonian complexity [17, 42] is to versal dynamics for a set of commuting Hamiltonians, pro- study the computational power of physical models described jected into a particular subspace. As often in Hamiltonian by local Hamiltonians, the intricate properties of their dynam- complexity, there are two views of this task, a static and a ics and their eigenstates, as well as to understand the com- dynamic one. Our goal is to uncover in which situations putational complexity of determining these properties. Many pinning-induced effective interaction terms (weighted sums of Hamiltonians are known to be universal for quantum compu- the original restricted terms) lead to an increase in complexity, tation [13], while others are thought to be much simpler, but or state preparation power. In both approaches, we prove sev- still hard to investigate classically [7] or even efficiently sim- eral results complementing what we know about the hardness ulable by classical computation [27]. There is a long history of problems without the special control. of searching for the simplest possible, closest to realistically First, statically, we ask about the difficulty of finding the and efficiently implementable, and robustly controllable inter- properties of low-energy states of pinned Hamiltonians. We action with universal dynamics for quantum computation with pin a qubit by an external prescription and show that deter- local Hamiltonians. Restrictions on the type and strength of mining the lowest energy in the pinned subspace is QMA- interactions, locality, and geometrical restrictions have been complete for a variety of restricted classes: commuting, sto- investigated, e.g., in Refs. [13, 20, 26, 37, 39, 40]. Thinking quastic, Markov, and permutation Hamiltonians. With this we about universality for computation often comes hand in hand wish to shed light on the complexity of these problems with- with asking complexity questions such as identifying the hard- out pinning, which we believe to be weaker. One of these is ness of determining the properties of the eigenstates of these the currently actively investigated Commuting Local Hamil- Hamiltonians. tonian problem, which can be NP-complete, or have ground Looking at this from a quantum control theory viewpoint states with topological order [3, 44]. At the same time, we provides us with an interesting observation. An extra level of know that the ground state connectivity problem for commut- control over a subsystem can result in a boost in state gener- ing Hamiltonians is QCMA-complete [19]. Another is the Sto- ation possibilities, or the difficulty of complexity questions. quastic Local Hamiltonian Problem, whose complexity is in We have seen this with the DQC1 (“one clean qubit”) model the class StoqMA which contains NP and MA, but is strongly [30, 36], whose single fully initializable (clean) qubit gives believed weaker than QMA [7]. Taking the method of pinning rise to quantum advantage over classical computation. Simi- to the extreme, we finally show that it yields QMA-hardness larly, if one is allowed to use magic states, computing with a results for Hamiltonians that are as simple as permutation restricted set of universal gates such as [8] be- matrices. Some of our results on determining ground state arXiv:2001.03636v2 [quant-] 23 Oct 2020 comes universal for quantum computation. Effectively fixing energies of pinned problems complement the conclusions of parts of the system to a particular state using perturbation gad- Ref. [25] involving energy of the highest excited state. gets allowed us to build complex effective Hamiltonians from Second, dynamically, asking about the preparation power simpler ones [24]. It has also been shown that a Zeno-effect of evolution with restricted time-independent Hamiltonians measurement of a small subsystem can grant universal power combined with Zeno pinning of a qubit, we find connections to a non-universal set of commuting gates [10]. to previous work on Hamiltonian purification [10, 41], show- In this work, we investigate the computational potential of- ing that the quantum Zeno-effect can drive efficient univer- fered by controlling a small subsystem. We focus on a spe- sal quantum computation in several restricted settings. This cific type of control called pinning – fixing the state of a includes, in particular, commuting Hamiltonians. Thanks to the details of the constructions, our results carry time and space requirements/guarantees from universal evolution mod- els with unrestricted Hamiltonians. ∗ Corresponding author: [email protected] Third, we find an application of pinning for the Ground 2

State Connectivity (GSCON) problem [18] and its variants II. PINNED LOCAL HAMILTONIANS: A COMPLEXITY with restricted types of terms. Specifically, we prove that VIEWPOINT GSCON with stoquastic Hamiltonians is QCMA-complete, complementing a similar recent result on Commuting GSCON A. Local Hamiltonians and states with fixed qubits [19]. In QMA, a verifier asks for a witness of the form |ψi, to which she adds a few auxiliary qubits and verifies it with a Finally, we note that there are strong limits to the pinning V . Does anything change, if she demands technique. First, dimensionality arguments from Ref. [41] that the witness must have a few qubits that are pinned to some mean a necessary increase in the size of the purified system fixed state? No, as the verifier can ask for but the pinned in which interactions are restricted. Second, we encounter qubits of the witness, supply those pinned qubits on her own, questions regarding locality of the required terms. Note that and verify the whole state as before. pinning does not allow us to create multiplicative effective Rather straightforwardly, we can show that problems in the terms, as perturbative gadgets do – creating effective 3-local class QMA can be verified using Pinned QMA and vice versa, terms from 2-local ones. We do not know if it is possible so that Pinned QMA = QMA. If we ask for a pinned proof of 0 to build gadgets for effective k + 1 local interactions from the form |ψ i = |ψi|0i, with one pinned qubit, the extra de- k-local Hamiltonian terms with the help of pinning, for our mand does not increase the complexity of the problem. If the 0 0 commuting or stoquastic settings. Many such questions with verifier that asks for |ψ i is V , the same thing can be verified low locality thus remain open. in QMA with a modified circuit V which adds one more aux- iliary system that stores a check of whether the pinned qubit is really |0i, and then does the verification V 0, accepting only if both are accepted. Thus, Pinned QMA can be verified in Doing something special on a single additional qubit is not QMA. On the other hand, for any QMA verifier circuit W 0 that new. Besides Ref. [19], where the idea has been exploited to demands a witness |φ0i, there exists a pinned version, which show that the GSCON problem is QCMA-complete already demands a witness |φi = |φ0i|0i with one extra qubit, and for commuting Hamiltonians, Jordan, Gosset and Love [25] whose verifier circuit W simply disregards the pinned qubit have used techniques tracing back to Ref. [22] to get rid of and verifies only the |φ0i part with W 0. varying signs of matrix elements by increasing the system size However, things are not quite as straightforward when in- and replacing positive 1’s by 2×2-identity matrices and neg- stead of QMA witnesses we start pinning qubits of low energy ative 1’s by the Pauli X matrices. They prove universality of states for the Local Hamiltonian problem. Let us consider the adiabatic quantum computation in an excited state of a Sto- QMA-complete problem Local Hamiltonian (LH), and inves- quastic Local Hamiltonian, instead of the usual ground state tigate the pinning requirement. Imagine we look at a Hamilto- computation, by splitting the Hilbert space into two, depend- nian H0, and ask if there exists a low energy state of the form ing on the state of an auxiliary qubit. Moreover, adding a sto- |ψ0i = |ψi|0i. We call this problem Pinned LH. quastic term effectively pinning this auxiliary into a state that results in a high energy, they showed QMA-completeness of Definition 1 (The p-Pinned k-Local Hamiltonian Problem). understanding energy bounds for the highest excited energy of Consider a k-local Hamiltonian H for a system of size n, a a Stoquastic Local Hamiltonian. Stoquastic Local Hamiltoni- p-qubit state vector |φi, with p = poly(n) and two energy ans are those local Hamiltonians whose matrix elements bounds b, a, such that b − a ≥ 1/poly(n). You are promised in the standard basis satisfy the condition that all off-diagonal that either: matrix elements are real and non-positive [6,9, 21, 29, 35]. YES There exists an n−p qubit state vector |ψi, such that the Next, they also show QMA hardness of bounding the lowest energy of the n-qubit state vector |ψi|φi with respect to energy of doubly stochastic (Markov) matrices, and QMA 1 H is at most a, or hardness of the Stochastic 6-SAT problem (deciding whether a sum of stochastic matrices is frustration-free or not). NO for any state vector |ψi, the energy of the n-qubit state vector |ψi|φi with respect to H is at least b. Decide, which is the case. This work is structured as follows: First, in SectionII, we show that several restricted versions of the Pinned Local We will prove the following theorem: Hamiltonian problem are QMA-complete, in particular, com- Theorem 2 (QMA-completeness of the Pinned k-Local muting, stoquastic and permutation Hamiltonians. In Sec- Hamiltonian Problem). The Pinned k-Local Hamiltonian tion III we then turn to the dynamical problem of univer- Problem is QMA-complete. sal time evolution, showing that the Zeno-pinned time evo- lution under both commuting and stoquastic Hamiltonians is Proof. First, on the one hand, Pinned LH is no easier than LH, complete for universal quantum computations. Finally, in because for any local Hamiltonian H, we can choose choose SectionIV, we prove that the stoquastic GSCON problem |φi = |0i and set is QCMA-complete and discuss the free fermionic GSCON 0 problem. H = H ⊗ I. (1) 3

There exists a low-energy state of H0 of the form |ψi|0i if producing and only if there exists a low-energy state vector |ψi of H. 1  p  Thus, Pinned LH is QMA-hard as solving it allows one to E ≥ c + b − (c − b)2 + (2d)2 . (10) solve the LH Problem. On the other hand, observe that Pinned S 2 LH belongs to QMA. We can set up a quantum verifier that receives the witness |ψi, adds its own single-qubit state vec- Let us now set tor |φi, and then tests whether the state vector |ψi|φi has low 1  (2d)2  enough energy for the Pinned Local Hamiltonian G0. In sum- c = b + a + , (11) 2 b − a mary, Pinned LH is QMA-complete. This could be the end of the proof. However, one might i.e., desire more details in order to understand how to translate the energy bounds between these problems. We can explic- b + a  2d  ∆ = c + d = + d + 1 = poly(n). (12) itly set up the LH problem to contain Pinned LH for example 2 b − a as follows. Let us construct a Local Hamiltonian G, which 0 has a low-energy state if and only if a Pinned LH G has a With basic algebra, recalling b > a, we can show that this low-energy state vector of the form |ψi|φi. Without loss of satisfies c − p(c − b)2 + (2d)2 ≥ a, and thus generality, we can again take |φi = |0i, by a local basis trans- formation on the operators acting on the last qubit. a + b Let us then set up a Local Hamiltonian G retaining the prop- ES ≥ , (13) 2 erties of a pinned G0 by penalizing the additional qubit with energy ∆ > 0 if it is not in the desired pinned state vector |0i, which means in the NO instances, the ground state energy will be at least (a + b)/2, which is at least an inverse polyno- 0 G = G + ∆ I ⊗ |1ih1|. (2) mial above the lower bound a in the YES instances. Together with (3), this means we have translated the original problem’s If there exists a state vector of the form |ψi|0i for the Pinned energy bounds to a0 = a and b0 = (a + b)/2, halving the Local Hamiltonian G0 with energy E ≤ a, the same state ψ,0 promise gap of the original Pinned LH. will also have a “low” energy for the local Hamiltonian G, 

h0|hψ|G|ψi|0i ≤ a. (3) Therefore, we have not really changed the complexity of the general local Hamiltonian problem by the pinning require- On the other hand, if it is the case that any state vector of ment. However, the situation surprisingly changes when we the form |ψi|0i has energy at least Eψ,0 ≥ b, then taking a start thinking about Hamiltonians whose terms come from a general state vector, restricted class, as we will show in the following sections.

|Si = (cos ϕ)|ψ0i|0i + (sin ϕ)|ψ1i|1i, (4) B. Pinned Commuting Local Hamiltonian we can show that the ground state energy of the local Hamil- tonian G obeys Pinning a qubit effectively projects into a subspace of the entire Hilbert space. When the original Hamiltonian comes ES = hS|G|Si (5) 2 0 with some restrictions, these may be lifted after this projec- = (cos ϕ)h0|hψ0|G |ψ0i|0i tion. Here and in the following sections, we investigate such 2 0 + (sin ϕ)(h1|hψ1|G |ψ1i|1i + ∆) cases. First, we claim that pinning a qubit for a commuting lo- 0 cal Hamiltonian and asking about the lowest possible energy + (cos ϕ sin ϕ)(h1|hψ1|G |ψ0i|0i + c.c.) of such a state is as difficult as asking about the ground state 2 2 ≥ b cos ϕ + ∆ sin ϕ (6) energy of a generic local Hamiltonian. 2 0 + (sin ϕ)h1|hψ1|G |ψ1i|1i Note that the complexity of the original (unpinned) Com- 0 muting Local Hamiltonian problem is an open question. The + (sin 2ϕ)Re [h1|hψ1|G |ψ0i|0i] restriction to commuting terms suggests the problem is not 2 0 2 0 ≥ b cos ϕ + (∆ − kG k) sin ϕ − sin 2ϕ kG k . (7) very different from classical. Schuch has showed that this 0 0 problem is in NP for plaquette (4-local) interaction terms Let us label c := ∆ − kG k and d := kG k to write on a square lattice of qubits [44], i.e. there exist classical b c proofs that such Hamiltonians have energy lower than some ES ≥ (1 + cos 2ϕ) + (1 − cos 2ϕ) − d sin 2ϕ. (8) bound. This result has been expanded and improved in work 2 2 by Aharonov et al. [2,3]. Importantly, though, the complexity Assuming c − b > d > 0, it is easy to find that the extrema of of the problem is unknown for generic graphs, larger locality, this expression appear at and larger local dimension terms. Importantly, already quite simple commuting local Hamiltonians have ground states with 2d tan 2ϕ = , (9) topological order (e.g. the [28]), so the complexity c − b of finding the properties of their ground states could be much 4 harder. In particular, the commuting ground state connectiv- qubit systems, with promise bounds b, a. As the original ity problem about the structure of the ground state is QCMA- problem is QMA-hard for k = 2, we have thus proven complete. Note though, that this is likely lower than QMA- that 3-local Pinned Commuting Local Hamiltonian is QMA- completeness. This all motivates us to investigate Pinned complete.  Commuting LH. We will now prove our first result: Note that our construction is not geometrically local, as it Theorem 3 (QMA-completeness of the Pinned Commuting requires interaction with the pinned qubit for all original parti- 3-local Hamiltonian problem). The Pinned Commuting 3- cles. We leave the possibility of geometric locality as an open local Hamiltonian problem is QMA-complete. question. The Pinned Commuting k-local Hamiltonian problem is defined analogously to Definition1, with an additional con- dition: the Hamiltonian’s terms commute with each other. Let C. Pinned Stoquastic Local Hamiltonian us prove it is QMA-complete. Proof. First, note that the Pinned Commuting k-local Hamil- Let us look at another restricted class – stoquastic Hamilto- tonian problem is in QMA, just as Pinned LH is. The harder nians with non-positive off-diagonal terms. For such Hamilto- direction is to show that commuting terms plus pinning can nians an important obstacle to classical simulation via Quan- result in complexity equal to the case of unrestricted local tum Monte Carlo – the sign problem – does not arise [34]. Hamiltonians. Thanks to Ref. [5], we know that the 2-local The local Hamiltonian for stoquastic Hamiltonians defines the Hamiltonian problem made from Z, X, ZZ, and XX terms StoqMA [6], which is believed to be strictly is QMA-complete. Let us take such a Hamiltonian and split smaller than QMA for the above reason. In particular, stoquas- it into two groups, one made from ZZ and Z terms, and the tic Hamiltonians are not thought to be universal for quantum other made from XX and X terms. The terms within each computing. What happens when we pin some of the qubits of P P such Hamiltonians? We show the following. group commute with each other. Let H = i Ai + j Bj be such a non-commuting k-local Hamiltonian, where in the Theorem 4 (QMA-completeness of the Pinned Stoquastic group A = P A , all the A commute with each other, and i i i 3-Local Hamiltonian problem). The Pinned Stoquastic 3- in B = P B , all the terms B commute with each other. j j j Local Hamiltonian problem is QMA-complete. Assume the Local Hamiltonian promise problem for this H has energy bounds b and a. Let us now add another qubit to A different viewpoint on this problem is given in Ref. [25], the system, and modify the terms to where the authors show universality of adiabatic evolution in the highest excited state of a stoquastic Hamiltonian, and the 0 1 A = Ai ⊗ (I + X) = Ai ⊗ |+ih+|n+1, (14) QMA hardness of lower bounding the highest energy of such i 2 n+1 1 a Hamiltonian. B0 = B ⊗ (I − X) = B ⊗ |−ih−| , j j 2 n+1 j n+1 Proof. As in the proof of Theorem3, we start with observ- similarly to the approach taken in Ref. [19]. These terms ing that Pinned stoquastic k-local Hamiltonian is in QMA, be- form a fully commuting, (k + 1)-local Hamiltonian H0 = cause Pinned LH is in QMA. We will now show that looking P 0 P 0 i Ai + j Bj. How much power would we have if we at the ground state energy of a Hamiltonian with stoquastic could figure out whether H0 has a low-energy state vector of terms with pinning a qubit results is as hard as for a general the form |ψi|0i? Observe on the one hand that when we pin local Hamiltonian. the last qubit to the state vector |0i, the expectation values of Let us start with an instance of the QMA-complete problem the Ai’s and Bj’s become Local Hamiltonian. For each such Hamiltonian H, we can write another using only stoquastic terms, in order to deal with 1 0 H h0|hψ|Ai|ψi|0i = hψ|Ai|ψi, (15) possible positive off-diagonal elements in . For this, we will 2 divide H = Oˆ + Pˆ into local terms Oˆ which are diagonal or 0 1 ˆ h0|hψ|Bj|ψi|0i = hψ|Bj|ψi. (16) have negative off-diagonal elements, and local terms P with 2 positive off-diagonal elements. Let us replace the latter with Thus, if the original k-local H has a ground state vector |ψi stoquastic terms as follows. First,√ add an extra qubit q in a with energy a, the state vector |ψi|0i will have energy a/2 state vector |−i = (|0i − |1i) / 2 to the system. Second, 0 0 for the new commuting Hamiltonian H , as h0|hψ|H |ψi|0i = modify each term Pˆ by attaching the operator Xq and change 1 0 2 hψ|H|ψi. On the other hand, if the energy of any state vector its sign, generating a new, stoquastic Hamiltonian H = Oˆ ⊗ |ψi for the Hamiltonian H is at least b, the energy of any state I − Pˆ ⊗ X . When we then look at state vectors of the form 0 q vector |ψi|0i for the new commuting Hamiltonian H is at |φi|−i, the expectation values of the modified Hamiltonian least b/2. will be Therefore, if one could solve a Pinned Commuting (k + 1)- Local Hamiltonian n + 1 0 problem on qubits, with promise h−|hφ|H |φi|−iq = h−|hφ|Oˆ ⊗ I − Pˆ ⊗ Xq|φi|−iq b , a , one could use this to solve a k-Local Hamiltonian prob- 2 2 ˆ ˆ lem (made from two commuting groups of terms) on an n = hφ|O + P |φi = hφ|H|φi. (17) 5

The expectation value of a pinned state vector |φi|−i for the Theorem 5 (QMA-completeness of the Pinned Local Per- stoquastic H0 is the same as for the state vector |φi and the mutation Hamiltonian). Pinned Local Permutation Hamilto- original Hamiltonian H. nian is QMA-complete, with a logarithmic number of pinned In more detail, let us start with the QMA-complete 2-local qubits. Hamiltonian made from terms X,Z,X⊗X,Z⊗Z, and X⊗Z [5]. First, we will change each term of the X type with a Note that (dynamical) universality for quantum computa- positive prefactor xa > 0 into tion with 0/1 matrices has been previously demonstrated for example in the PromiseBQP-rewriting problem of Wocjan xaXa 7→ −xaXa ⊗ Xq, (18) and Janzing [23], or the universal computation by quantum walk construction of Childs et al. [12]. which is stoquastic. When we pin the qubit q in the state vector |−i , the expectation value of the new term in the q Proof. One direction of Theorem5 is easy – pinned local per- state vector |φi|−i will be simply x hφ|X |φi, thanks to q a a mutation Hamiltonian is obviously in QMA. The more diffi- h−|X |−i = −1. We can deal with the terms of the type q cult part is again to construct QMA-hard instances of pinned XX with a positive prefactor just as easily. Next, we will 0/1 Hamiltonian. First, we will take a target Hamiltonian look at the terms X ⊗ Z in H, whose off-diagonal terms made from , and replace them by 0/1 matrices have a varying sign. Because we can rewrite X ⊗ Z = on a larger Hilbert space, with a technique similar to those of X⊗|0ih0|−X⊗|1ih1|, assuming x > 0, the corresponding a,b Ref. [25], where it has been used to build QMA-hard instances terms in H0 will be of stochastic matrices. Second, we will utilize pinning to gen- erate the desired real-valued prefactors for the permutation, xa,bXa ⊗ Zb (19) and thus also the effective original Pauli terms. 7→ −x X ⊗ (|0ih0| ⊗ X + |1ih1| ⊗ I ) , a,b a b q q Consider an instance of the QMA-complete, 2-local Hamil- −xa,bXa ⊗ Zb (20) tonian problem with a Hamiltonian H made from X, Z, XX 7→ −xa,bXa ⊗ (|0ih0|b ⊗ Iq + |1ih1| ⊗ Xq) . and ZZ terms, as in SectionIIB, with real-valued prefactors. Let us deal with Pauli terms first, and consider the prefactors Observe that the modified terms are stoquastic, with only neg- later. The X and XX terms already are permutation matrices. ative off-diagonal elements. For the Z and ZZ terms, we will add an auxiliary qubit z, and 0 Consider now the new stoquastic 3-local Hamiltonian H transform the interactions as and ask whether its low-energy vectors can have the form

|φi|−i. On the one hand, if the original H has a ground Z 7→ |0ih0| ⊗ Iz + |1ih1| ⊗ Xz, (21) state vector |φi with energy a, the state vector |φi|−i will Z ⊗ Z 7→ (|0, 0ih0, 0| + |1, 1ih1, 1|) ⊗ I (22) have energy a for the new stoquastic Hamiltonian H0. On the z other hand, if the energy of any state vector |φi for the Hamil- + (|0, 1ih0, 1| + |1, 0ih1, 0|) ⊗ Xz, tonian H is at least b, the energy of any state vector of the form |φi|−i is at least b for the new commuting Hamiltonian generating 2-local and 3-local permutation matrices, made H0 Local Hamiltonian from 0/1 elements. This results in a permutation Hamilto- . Therefore, we have turned a problem 0 with promise parameters a, b, into a Pinned Stoquastic Local nian H . When we pin the aixuliary qubit z in the state vector Hamiltonian with the same promise, with a doubled Hilbert |−i, we can effectively generate the original Z and ZZ (and space (adding a qubit), and stoquastic terms that have a local- of course X and XX) terms as we did for stoquastic Hamil- ity increased by 1. Solving Pinned Stoquastic LH is thus at tonians. Second, we want to generate real-valued prefactors for the least as hard as LH, and thus QMA-complete.  effective Pauli terms using permutation Hamiltonians. This is Note that in the proof we provided, the type of the terms in straightforward with the help of pinning, once we add and pin H0 is different from H, as we were only interested in making several auxiliary systems. In the definition of Pinned Local them stoquastic, not keeping their form. It remains open to an- Hamiltonian, we allow for pinning of up to a polynomial num- alyze what is the hardness of Pinned Stoquastic Hamiltonian ber of qubits. In the problems considered so far, we pinned a with restricted form (e.g., only XXX, ZZZ) or locality below single qubit. Here, we will use a logarithmic number of such 3. After showcasing the pinning technique in two examples, auxiliary systems. we will continue exploring how far it takes us, applying it to Let us start with a system described by the Hamiltonian 0 P simpler and simpler original Hamiltonians. H = i Pi built in the previous step as a sum of permuta- tion matrices, with only 0, 1 elements, with a single 1 in each row and column. We will show how to add Q + 1 qubits and D. Pinned Permutation Hamiltonians interactions to form H00. Pinning the Q new auxiliary systems to a specific product subspace S, will then allow us to effec- 00 The possibilities opened in the previous sections motivate tively investigate the target Hamiltonian H = ΠS H ΠS with us to go further and design a classically looking problem about the desired form, up to precision 2−Q for its terms. This pre- 0/1 permutation matrices that will still be QMA-complete. cision comes from the possibility of imprecisions of the orig- This is a further restriction on stoquastic Hamiltonians. We inal Local Hamiltonian problem. If the original problem was claim the following. given precisely, but with an inverse polynomial promise gap, 6 allowing for an inverse-polynomial imprecision in the Hamil- restricted set of interactions (or unitaries). We will consider tonian’s elements simply shrinks the promise gap, if we con- constantly measuring one qubit in a particular basis, pinning sider a large enough Q, which is however still logarithmic in it via the Zeno effect to a particular state. Note that this is n. different from postselection. There, one is allowed to choose Recall our target effective Hamiltonian H has general real a particular result of a measurement of a subsystem without prefactors for its Pauli terms. Let us consider the terms from regard of the result’s (im)probability. This would give one the permutation Hamiltonian H0 from the first step. Imagine immense computational power [1], as postselected quantum we want the term Oˆ to have a prefactor 0 < x < 1. We will computation has the power of PP, much larger than NP. Pin- decompose x into binary, up to some precision Q, as ning does not allow us to choose a measurement result freely. Instead, we must rely on the Zeno effect to give us a high prob- Q ability of the desired projection. Pinning is thus applicable in X xj x = , (23) 2j practice, unlike the theoretical concept of postselection. j=1 With frequent projective measurement, we effectively get access to a specific state of a qubit, and thus a specific sub- with xj ∈ {0, 1}. For each nonzero xj, we will pin an auxil- space of the whole Hilbert space. We will show that the dy- iary qubit qj to namics of restricted Hamiltonians in this chosen subspace can 1 result in universal dynamics. In the circuit model, we know |α i = cos α |0i + sin α |1i, sin 2α = , (24) that access to specific states can greatly enhance the power of j j j j 2j a restricted model. For example, a source of magic states is 0 enough to turn computation with Clifford gates into a univer- with a new term Oˆ ⊗ Xq in H for each nonzero xj, in or- j sal quantum computation [8]. Following a similar strategy as der that hψ|hα |Oˆ ⊗ X |ψi|α i = hψ|Oˆ|ψi/2j. Pinning the j qj j in the previous section, we will now show how to get univer- Q auxiliary qubits to their respective state vectors |α i, alto- j sal quantum computation out of evolution with a restricted set gether they become an effective Hamiltonian of (e.g. commuting) Hamiltonians together with a fixed Pauli

 Q  basis measurement of a single qubit. X xj Oˆ (25)  2j  j=1 A. Warm-up: evolution with pinned stoquastic Hamiltonians on the n qubits of the system. Second, to generate effective negative prefactors, we use the standard trick from before, We will start with a simple example of applying pinned evo- adding the auxiliary qubit q0 pinned in the state vector |−i, lution to stoquastic Hamiltonians. We know that evolution ˆ with stoquastic Hamiltonian is already universal for quantum and an interaction of the form O ⊗ Xq0 to the desired terms. Let us summarize. Our target Hamiltonian H acting on n computation, as shown by Childs et al. [11]. However, our qubits has M Pauli terms with real prefactors and locality at pinned construction has its own merits, even over later de- most 2. In step 1, we built an n + 1 qubit permutation Hamil- velopments [12], in terms of space/time requirements. More- tonian H0 with locality at most 3, which did not yet include over, it will be useful in SectionIVA, where we will use it the desired real prefactors. In step 2, we constructed the fi- for the proof of QCMA-hardness of the Stoquastic GSCON nal permutation Hamiltonian H00 which works on n + Q + 2 problem. Note also that Fujii has shown how adding local qubits, and has at most 2M × (Q + 1) terms, with locality at measurements to adiabatic evolution with stoquastic Hamil- tonians (stoqAQC) results in universality (for adaptive mea- most 5. We pinned the auxiliary qubits z and q0 into the state surements) or quantum advantage (if using non-adaptive mea- vector |−i, and the auxiliary qubits q1, . . . , qQ into the states (24). Determining the lowest energy of the pinned 5-local per- surements) [16]. Again, what we do here is more efficient, mutation Hamiltonian H00, with Q + 2 pinned qubits, is thus requires smaller locality and easier control. QMA-hard, as it implies determining the ground state energy Let us then look at a system with a stoquastic Hamiltonian of the target local Hamiltonian H, an instance of the QMA- H0 = A ⊗ I + B ⊗ X , (26) complete problem Local Hamiltonian. Therefore, the Pinned q q permutation Hamiltonian problem is QMA-hard, as well as made from two groups of local, stoquastic terms A and B, QMA-complete.  with no positive off-diagonal entries. Furthermore, we de- mand B to be entirely off-diagonal. The terms B⊗Xq include an interaction with an auxiliary qubit q, similarly to (19). We III. A DYNAMICAL VIEW OF PINNING can now show that pinned evolution with time-independent, local stoquastic Hamiltonians is universal for BQP as follows. In the previous section, we looked at how pinning can con- We initialize the auxiliary qubit q as |−i, and measure it in tribute to the complexity of determining the static properties the X basis often enough. This likely pins the auxiliary qubit of local Hamiltonians – the bounds on the energies of states to the state vector |−i. Meanwhile, the system evolves with from the pinned subspace. We now turn to a dynamical ques- H0. This results in a particular effective evolution. Let us cut tion, asking what pinning can contribute when applied to an the time evolution into small steps of size δ → 0. The evo- evolution with a local, time-independent Hamiltonian with a lution can be approximated as alternating the evolution e−iδH 7 with a projection of the last qubit onto the state vector |−i. It the auxiliary qubit q in the state vector |0i, as in Vaidman’s will be helpful to express bomb-testing procedure [15] in its circuit setting [31]. This is the Zeno effect, explained in detail e.g., in Ref. [38], where −iδH −iδA −iδB⊗Xq h−|e |ψi|−iq ≈ h−|e e |ψi|−iq we also find that the probability of a “bad” projection (a flip 2 ≈ h−|(1 − iδA − iδB ⊗ Xq)|ψi|−iq of the |0i to |1i scales as O δ , and can be made arbitrar- −1 = 1 − iδ(A − B)|ψi ≈ e−iδ(A−B)|ψi, (27) ily small even after O δ repetitions. On the other hand, what is the effective evolution of the rest of the system? Let valid up to first order in δ. This allows us to effectively evolve us calculate the state vector |ψi with the general, non-stoquastic Hamilto- −iδH0 −i2δA⊗|+ih+| −i2δB⊗|−ih−| nian H = A − B. e |ψi|0i = e e |ψi|0i (30) Moreover, because the last qubit is in an eigenstate of Xq, ≈ (I − i2δA ⊗ |+ih+| − i2δB ⊗ |−ih−|) |ψi|0i (31) it never gets flipped into the state vector |+i. Thus, taking i2δ δ = t/N, with N → ∞, we can confidently say that ≈ |ψi|0i − √ (A|ψi|+i + B|ψi|−i) (32) 2

 N  = (I − iδ(A + B)) |ψi|0i − iδ(A − B)|ψi|1i (33) Y  −iδ(A+B⊗X) |ψ(t)iPE =  P− e  |ψi ≈ e−iδ(A+B)|ψi|0i − iδ(A − B)|ψi|1i, (34) j=1 = e−it(A−B)|ψi + |δi, (28) correct up to order δ. Therefore, when we now measure the auxiliary qubit, we will get the result 0 and obtain the 2 where |δi is an error state vector with norm of order at most state e−iδ(A+B)|ψi, with probability 1 − O((δ kA − Bk) ). δ = t/N, i.e., going to zero as N → ∞. Therefore, we can Moreover, the state can also contain an error vector with simulate evolution with a time-independent, non-stoquastic norm O((δ kA + Bk)2), as the evolution (30) with commut- local Hamiltonian using evolution according to a stoquas- ing terms cannot produce mixed terms such as AB, while (34) tic local Hamiltonian (with locality increased by 1) and pin- does include them in its series expansion. ning. This is universal for quantum computation (BQP), when What happens when we repeat this evolve-measure pro- we recall various standard constructions for universal quan- cedure N = t/δ times? We end up with the state vector tum computation by evolution with a time-independent, non- e−it(A+B)|ψi, with an error vector of norm O(tδ kA + Bk2), stoquastic local Hamiltonian, e.g., Ref. [37]. while the probability that all the N measurements of the pinned auxiliary qubit result in |0i is lower bounded by 1 − O(tδ kA − Bk2). Therefore, we can simulate evolution B. Pinned evolution with commuting Hamiltonians with unrestricted (non-commuting) Hamiltonians using com- muting Hamiltonians and pinning. Starting with a universal Let us now turn to our main result about pinned dynam- local Hamiltonian built from two groups of commuting terms ics. We will investigate what kind of evolution we can achieve as in SectionIIB, this directly translates into the statement of with pinned commuting local Hamiltonians. A similar ques- the theorem: pinned evolution with commuting local Hamil- tion has been posed in the context of Hamiltonian purifica- tonians is universal for quantum computation.  tion, with an emphasis on obtaining an universal algebra [10]. Here, we will show how to efficiently simulate evolution with State preparation with a universal, 2-local, non-commuting a non-commuting Hamiltonian H = A + B, made from two construction [37] that has O() gates in a circuit can thus be groups of terms that commute within the group. For this, we efficiently simulated with low error by time O(L) evolution will construct a Hamiltonian with 3-local, commuting terms, and frequent measurement of a single qubit. 0 H = 2A ⊗ |+ih+|q + 2B ⊗ |−ih−|q (29) all of whose terms commute, by adding an auxiliary qubit q as IV. GROUND STATE CONNECTIVITY in (14). Let us now analyze what happens when we alternate computational basis measurements on the last qubit, initial- 0 Our original motivation for exploring pinning was to un- ized as |0i, with evolution according to H . We will prove the derstand better the variants of Gharibian and Sikora’s Ground following. State Connectivity (GSCON) problem [18]. It asks about the Theorem 6 (Universality of commuting Pinned Evolution). possibility of traversing the low-energy subspace of a local Pinned evolution with time-independent, local commuting Hamiltonian from one specific ground state to another, using Hamiltonians is universal for BQP. local unitary transformations. Gosset, Mehta and Vidick [19] have shown that the problem remains QCMA complete even Proof. Let us look at a short time interval δ = t/N, with if only commuting Hamiltonians are used. In their proof, they N → ∞. The pinned evolution of the system will be well ap- use a trick similar to pinning – combining the original Hamil- 0 proximated by the evolution e−iδH according to H0 for time tonian’s terms with projections on auxiliary qubits to make δ, and then a measurement in the computational basis. This the terms commute. Then they demand that the initial and fi- repeated measurement should on the one hand effectively pin nal ground state have a few qubits in a specific state – which 8 means that the original non-commuting Hamiltonian’s terms (a) (Intermediate state obtains high energy) There ex- are effectively applied. Moreover, this has to be combined ists i ∈ [m] and an intermediate state vector with the impossibility of a simple flip of this state without a |ψii := Ui ··· U2U1|ψi, such that hψi|H|ψii ≥ computation being verified first. Nevertheless, it helped us re- η2, or alize that the GSCON formulation allows one to essentially fix (b) (Final state far from target state) some part of the ground state, adding extra power to restricted kU ··· U |ψi − |φik ≥ η , forms of Hamiltonians. m 1 4 Therefore, using techniques similar to Ref. [19], hardness then output NO. results for pinned local Hamiltonians should be translatable to hardness of GSCON for similarly restricted Hamiltonians. For There is not that much that we need change in the proof example, we will be able to show QCMA-hardness of GSCON of Theorem 6 in Ref. [19], when we want to build a generic for stoquastic Hamiltonians, building on Ref. [19] and the effective Hamiltonian from stoquastic instead of commuting construction from Section III A. Moreover, in this context we terms, using “pinning” thanks to a restriction on the initial will also provide some evidence into the free-fermionic vari- and final states, as well as the form of the Hamiltonian that ant of GSCON, to be further developed in future work. we construct.

Theorem 8 (QCMA-completeness of the Stoquastic Ground A. Stoquastic GSCON State Connectivity Problem). The Stoquastic Ground State Connectivity Problem is QCMA-complete.

First, we will show how to build on the proof that the Proof. It is straightforward to see that the Stoquastic GSCON Ground State Connectivity (GSCON) problem is QCMA- is in QCMA, with a witness encoding the sequence of uni- complete for commuting Hamiltonians, as well as on uni- taries, verifiable by a quantum computation. For the other versality of pinned stoquastic LH, and prove that Stoquastic direction, we are directly inspired by the proof of QCMA- GSCON is QCMA-complete. The statement of the problem is completeness of Commuting GSCON [19]. There, the au- identical to the Commuting GSCON problem in Ref. [19], the thors split a target generic (non-commuting) local Hamilto- only difference being the replacement of the word “commut- nian G = A + B into two groups of local commuting terms, ing” by “stoquastic”. We thus have: add two 3-qubit auxiliary registers, and set up the commuting Hamiltonian Definition 7 (Stoquastic Ground State Connectivity (H, k, η , η , η , η , ∆, l, m, U ,U )). 1 2 3 4 ψ φ A ⊗ ΠS ⊗ Π+ + B ⊗ ΠS ⊗ Π− + I ⊗ I ⊗ ΠS , (35) Input: P where ΠS projects onto S = span {|0, 0, 0i,√|1, 1, 1i}, 1. k-local Hamiltonian H = Hi with stoquastic terms i and Π± are projectors onto (|0, 0, 0i ± |1, 1, 1i) / 2. The (i.e. with no positive off-diagonal elements), satisfying QCMA-hard GSCON question concerns the possible low- kHik ≤ 1. energy traversal from the state vector |0i⊗n|1i⊗3|0i⊗3 to the state vector |0i⊗n|0i⊗3|0i⊗3 by 2-local operations. This is 2. η , η , η , η , ∆ ∈ , and integer m ≥ 0, such that 1 2 3 4 possible by using the first n-qubit register to prepare a low- η − η ≥ ∆ and η − η ≥ ∆. 2 1 4 3 energy witness for the Hamiltonian G = A + B. This ef- fectively “turns off” the first two terms in (35), allowing one 3. Polynomial size quantum circuits Uψ and Uφ gener- ating “starting” and “target” state vectors |ψi and |φi to flip the middle register to |1, 1, 1i by 2-local operations starting from the |0i⊗n state, respectively, satisfying without a high energy cost. Finally, one uncomputes the first register. Meanwhile, the last register stays “pinned” in hψ|H|ψi ≤ η1 and hφ|H|φi ≤ η1. |0, 0, 0i, making sure both groups of terms A and B are in Output: play and contribute significantly to the energy of the interme- diate states. For more details, see the proof of Theorem 6 of m 1. If there exists a sequence of l-local unitaries (Ui)i=1 ∈ Ref. [19]. U such that Let us then work out the stoquastic version of this. We start with an n-qubit register, and the target generic, non- (a) (Intermediate states remain in low energy space) stoquastic, 2-local, n-qubit Hamiltonian H made from ZZ, For all i ∈ [m] and intermediate states ZX, XX, Z and X terms. The Local Hamiltonian problem |ψii := Ui ··· U2U1|ψi, one has hψi|H|ψii ≤ η1, for this variant of H is QMA-complete. The GSCON problem and based on H is thus QCMA-complete. 00 (b) (Final state close to target state) We will construct a stoquastic GSCON Hamiltonian H kUm ··· U1|ψi − |φik ≤ η3, similarly to (35), with a few important differences. First, let us define two operators then output YES. 1 m Q = (Xq1 + Xq2 + Xq3 ) , (36) 2. If for all l-local sequences of unitaries (Ui)i=1, either: 3 9 an analogue of Xq from SectionIIC, effectively flipping the For soundness, one can directly follow [19] to show that sign when the auxiliary register is in the state vector |−i⊗3, no sequence of 2-local unitaries will satisfy well enough the and two conditions – end near enough the final state and stay low enough in energy throughout the sequence. The lower bound 3 1 R3 = I − (Xq1 Xq2 + Xq2 Xq2 + Xq1 Xq3 ) , (37) on the energy of the intermediate states if one is to end up 4 4 2 6 close to the final state is in this case η2 = Ω β /m , just as a 2-local, stoquastic operator equivalent to the projector onto in the proof of Soundness of Theorem 10 in Ref. [19], where the space orthogonal to the span of |−i⊗3 and |+i⊗3. hψ|H|ψi ≥ β is the bound in the NO case of the original LH Second, let us add a 3-qubit auxiliary register and combine problem and m is the number of unitaries in the sequence. 0 the original Hamiltonian H with the operator R3 as H = One has only to replace H ⊗ R3. Similarly to SectionIIC, we can split this local P = |0, 0, 0ih0, 0, 0| 7→ |−ih−|⊗3, (42) Hamiltonian H0 acting on n + 3 qubits into groups of local 0 ⊗3 terms H0 = Oˆ0 + Pˆ0, with non-positive off-diagonal terms Oˆ0 P1 = |1, 1, 1ih1, 1, 1| 7→ |+ih+| , (43) and a group of strictly off-diagonal local terms with positive and follow the proof.  elements Pˆ0. Finally, we combine the group Pˆ0 with the operator Q on Observe that in the NO case, to obtain soundness, an effi- the final auxiliary register, in order to ensure that −Pˆ0 ⊗ Q cient (poly-length) sequence of 2-local transformations keep- is stoquastic, with strictly negative off-diagonal elements, as ing the energy of intermediate states low enough simply could Pˆ0 ⊗ Q is a tensor product of two operators which each have not exist, and this was guaranteed by the lower bound from strictly positive off-diagonal elements and no diagonal ele- the Small Projection Lemma 8 [19]. Would this be also true ments. Altogether, we arrive at the local, stoquastic Hamil- in other settings besides history state preparation connected to tonian QCMA-complete problems? We ask this question about quan- tum memories, e.g., based on the toric code, in forthcoming 00 0 0 H = Oˆ ⊗ I − Pˆ ⊗ Q + I ⊗ R3. (38) work.

Observe that for the state vectors of the form |ψi|−i⊗3|−i⊗3 and |ψi|+i⊗3|−i⊗3, the expectation value of H00 is zero. B. Ground state connectivity for free fermions Meanwhile, when the middle register is in an X-basis state ⊗3 ⊗3 vector |x1, x2, x3i other than |−i or |+i , and the last In the context of studies of Majorana fermionic quantum register remains in |−i⊗3, the expectation value memories, variants of GSCON for free fermions are partic- ularly interesting [4, 33]. Here we provide insights that we ⊗3 00 ⊗3 hψ|hx1, x2, x3|h−| H |ψi|x1, x2, x3i|−i expect to be helpful in tackling this version of the problem 0 0 relevant when assessing Majorana fermionic quantum memo- = hψ|hx1, x2, x3|Oˆ + Pˆ |ψi|x1, x2, x3i (39) ries: we provide evidence that between any pair of low-energy = hψ|hx1, x2, x3|H ⊗ R3|ψi|x1, x2, x3i = hψ|H|ψi free-fermionic states, there exists a local free-fermionic cir- (40) cuit that interpolates between them within the low-energy sub- is equivalent to the expectation value of the original non- space. Before we get there, let us define the Free Fermionic stoquastic Hamiltonian H acting on |ψi, thanks to Ground State Connectivity Problem, though. Note also that our discussion of the free-fermionic problem does not rely on pinning, but complements our understanding of GSCON in a hx1, x2, x3|R3|x1, x2, x3i = 1. (41) practically relevant setting. The hard ground space traversal question we ask is then: De- Definition 9 (Free Fermionic Ground State Connectivity cide, if starting in the state vector |0i⊗n|−i⊗3|−i⊗3, one can (H, k, η1, η2, η3, η4, ∆, l, m, Uψ,Uφ)). traverse the low-energy subspace of H00 without energy above α (where this bound comes from the QCMA-complete LH 1. Input parameters: problem with energy bounds α and β) and at most η far from 3 (a) k-local free fermionic Hamiltonian H = P H the state |0i⊗n|+i⊗3|−i⊗3, using a sequence of 2-local uni- i i acting on n fermionic modes with each H being taries of length polynomial in n, or whether one must end at i supported on no more than k modes, satisfying least η4 far from the final state, or some of the intermediate kHik ≤ 1. states have energy at least η2? Showing completeness is straightforward with the follow- (b) η1, η2, η3, η4, ∆ ∈ R, and integer m ≥ 0, such ing sequence of transformations. Note the third register stays that f η2 − η1 ≥ ∆ and η4 − η3 ≥ ∆. in |−i⊗3 throughout the process. First, we prepare the low- (c) Polynomial size fermionic Gaussian quantum cir- energy witness for H in the first register. The energy is zero cuits Uψ and Uφ generating “starting” and “tar- during this process. Second, we flip the second register from get” fermionic Gaussian state vectors |ψi and |−i⊗3 to |+i⊗3, qubit by qubit. In this process, the energy of |φi (starting from the fermionic vacuum), respec- the states is at most α, thanks to (40). Finally, we uncompute tively, satisfying hψ|H|ψi ≤ η1 and hφ|H|φi ≤ the first register, keeping the energy zero. η1. 10

2. Output: depending on having even or odd parity. Turning to Hamilto- nians, energy expectation values are computed as (a) If there exists a sequence of l-local unitaries m (Ui)i=1 ∈ U supported on m modes each such hψ|H|ψi = tr(γh), (48) that T P i. (Intermediate states remain in low energy with h = −h . For a local Hamiltonian H = i Hi, each T space) For all i ∈ [m] and intermediate states of the terms Hi will correspond to a matrix hi = −hi with + |ψii := Ui ··· U2U1|ψi, one has khik ≤ 1 that is a zero matrix except a 2k × 2k block, since hψi|H|ψii ≤ η1, and each hi acts on k modes only. The Hamiltonian matrix h can ii. (Final state close to target state) without loss of generality be assumed to be 2 × 2 block diag- onal, as any special orthogonal transformation to bring it into kUm ··· U1|ψi − |φik ≤ η3, this form can be absorbed in the O of the initial covariance then output YES. matrix. The attainable energy expectations can be computed m (b) If for all l-local sequences of unitaries (Ui)i=1, from the reachable set either:  T i. (Intermediate state obtains high energy) P (Oγ0O ): O ∈ SO(2n) , (49) There exists i ∈ [m] and an intermedi- where P is the projection onto 2 × 2 block diagonal form. ate state vector |ψ i := U ··· U U |ψi, such i i 2 1 By virtue of the analog of the Schur-Horn theorem for skew- that hψ |H|ψ i ≥ η , or i i 2 symmetric matrices [32], it becomes clear that within both the ii. (Final state far from target state) even and the odd parity sectors, the reachable set are all 2 × kUm ··· U1|ψi − |φik ≥ η4, 2 skew-symmetric real block diagonal matrices for even and then output NO. odd parity, respectively. As a consequence of that, there is a parametrized curve t 7→ O(t) for t ∈ [0, 1] with O(t) ∈ Here, we do not assess the hardness of the Free Fermionic SO(2n) for all t so that GSCON problem. We conjecture that in contrast to the gen- eral case, in free fermions there will always exist a local low- T γ = O(0)γ0O(0) (50) energy path between any pair of low-energy quantum states. Conjecture 1 (Free Fermionic Ground State Connectivity). and For any free fermionic Hamiltonian H and any pair of low- ω = O(1)γ O(1)T (51) energy Gaussian fermionic states |ψi, |φi there exists a 2-local 0 finite Gaussian fermionic circuit interpolating between them so that such that all intermediate states satisfy the energy constraint. T We here provide evidence in favour of this conjecture. Let tr(O(t)γ0O(t) h) = (1 − t)tr(γh) + ttr(ωh). (52) us denote the fermionic covariance matrix of the initial state That is to say, one can linearly interpolate between the ini- |ψi γ vector with (in the conventions of Ref. [14]), and with tial and final energy values. One can then chop the linear ω |φi n the covariance matrix of the final state vector . For interpolation into a finite number N steps, each of which is 2n × 2n γ = −γT modes, this is a real matrix satisfying (as characterized by an orthogonal matrix in SO(2n) close in γT γ = I is the case for any covariance matrix) and (reflect- operator norm to the identity. What is more, following the ing purity). The application of Gaussian fermionic gates to special orthogonal fermionic analog of the decomposition of |ψ i = U ··· U U |ψi achieve i i 2 1 corresponds to a transforma- Ref. [43], this transformation can be exactly decomposed into tion a an O(n2) sized circuit of 2-local fermionic Gaussian quan- T T T tum gates that are also close to the identity. The so obtained γi := Oi ··· O2O1γO1 O2 ··· Oi (44) QO(n2)N discrete local fermionic circuit i=1 Oi therefore remains with Oi ∈ SO(2n) for all i, on the level of covariance matri- close to the continuous curve O(t) for all t ∈ [0, 1]. This ces. In the Free Fermionic Ground State Connecttivity Prob- implies that the energy along this circuit cannot deviate too lem, the initial covariance matrix can be written as much from the initial and final value. By increasing the value T N γ = Oγ0O (45) of we can push this deviation down arbitrarily far so as to satisfy the energy constraint throughout the path, providing with O ∈ SO(2n) and either evidence for our conjecture. We leave the details of this inter- n esting problem relevant for practical quantum memories with M  0 1  γ = (46) Majorana fermions for future work. 0 −1 0 j=1 or V. DISCUSSION  n−1   0 −1  M  0 1  γ = ⊕ (47) Pinning exemplifies the mathematical question of Hamilto- 0  1 0 −1 0  j=1 nian purification [41], which we looked at here in a variety of 11 contexts (commuting, stoquastic, permutation, and other re- dition in the Pinned Stoquastic LH problem. Similarly, we stricted classes of Hamiltonians). We have presented several added dynamical pinning based on repeated measurements in results in Hamiltonian complexity, raising questions about the Section III as an external resource, and not directly as a part of static (complexity) and dynamical (evolution and universality) the Hamiltonian. Third, it would be interesting to see whether implications of a special type of control on a small subsystem. pinning for some restricted models could result in intermedi- Let us now discuss a few observations. ate complexity (e.g., completeness for transverse Ising mod- First, quantum perturbation gadgets that have been used in els), as classified in Ref. [13]. Hamiltonian complexity for a long time ever since [26], are Fourth, as pinning fixes a particular value of a certain sub- also based on a form of pinning – effectively fixing part of a system, one naturally asks about its relationship to postselec- system into a subspace by providing a large energy penalty tion. What we propose in Section III is far from postselection. to the orthogonal subspace. They can result in an effective In our Zeno-effect constructions, the probability of even many Hamiltonian with multiplicatively combined, higher-locality successful projections tends to 1. Our results say that univer- terms, thanks to the form of the perturbative expansion of the sality can arise even from this small degree of practical con- Hamiltonian’s self-energy. On the other hand, pinning as we trol. On the other hand, postselection is about being able to view it here, is a geometrical restriction on a part of a system. postselect (“choose” the value of measurement results regard- First of all, it is not perturbative, and second, it can effectively less of the low probability of the outcome). It is known that generate only linear and not multiplicative combinations of this incredibly powerful ability would increase the computa- operators. Therefore, it does not allow one to combine oper- tional power immensely – e.g., postselected BQP becomes PP ators to increase the effective locality of terms, which pertur- [1]. bative gadgets are designed to do. On the contrary, we need Fifth, we hope that our investigation will shed light on and k + 1 local terms in a pinned Hamiltonian to get an effec- motivate further inquiries into the complexity of the variants tive k-local Hamiltonian. In particular, to show that Pinned of the original local Hamiltonian problems – stoquastic, com- Commuting 3-LH is QMA complete in SectionIIB, we have muting, or with other restrictions. turned a 2-Local Hamiltonian problem with promise b, a, into a pinned version with a doubled Hilbert space by adding a Finally, we hope that dynamical pinning based on extra qubit. Moreover, the newly formed up to 3-local and com- control (repeated measurements) of a single qubit, described muting terms have the form Z, X, ZX, XX, ZZ, ZZX, or in Section III, with a fixed interaction Hamiltonian of a re- XXX. However, is the increase in locality essential? The stricted form, could be readily implemented in today’s ex- complexity of Pinned 2-Local Commuting Hamiltonian re- perimental settings. It is also our hope that the present work mains open. Straightforward attempts mimicking perturbation can substantially contribute to the growing body of solutions gadgets to generate effective interactions with higher locality to problems in Hamiltonian complexity beyond assessing the do not work. Similarly, we have shown in SectionIIC that computational complexity of approximating ground state en- the Pinned Stoquastic 3-LH is QMA-complete. However, it ergies, signifying the richness of the field. remains open to figure out how hard the Pinned Stoquastic 2- LH problem is. One way to go could be to show that 2-LH with ±ZZ, −XX, ±X, ±Z terms is QMA-complete. Second, our reason for investigating pinning was its appli- VI. ACKNOWLEDGEMENTS cation to Hamiltonians with a restricted form. Could pin- ning be “forced” with such restricted terms? Sometimes, as We thank an anonymous referee for valuable comments to in the application to GSCON, there exist operators with the the early version of this paper, especially on universal evolu- desired form, which energetically penalize a subspace. For tion with stoquastic Hamiltonians. D. N. has received fund- example, in SectionIVA, we wrote down the stoquastic op- ing from the People Programme (Marie Curie Actions) EU’s erator (37) that works as a projector onto the complement of 7th Framework Programme under REA grant agreement No. |−i⊗3 and |+i⊗3, or in Ref. [19], where a 3-local projector 609427. His research has been further co-funded by the Slo- has the required form commuting with the rest of the Hamil- vak Academy of Sciences, as well as by the Slovak Research tonian. However, in other situations we can not do this. For and Development Agency grant QETWORK APVV-14-0878 example, we can not energetically prefer the state vector |−i and VEGA MAXAP 2/0173/17. D. H. and J. E. have been of a qubit by stoquastic terms, as that would imply QMA- supported by the ERC (TAQ), the Templeton Foundation, and completeness of the Stoquastic LH problem, which is consid- the DFG (EI 519/14-1, EI 519/15-1, CRC 183). M. S. thanks ered unlikely. Thus, we require pinning as an external con- the Alexander-von-Humboldt Foundation for support.

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