Analysis and limitations of modified circuit-to-Hamiltonian constructions

Feynman's circuit-to-Hamiltonian construction connects quantum computation and ground states of many-body quantum systems. Kitaev applied this construction to demonstrate QMA-completeness of the local Hamiltonian problem, and Aharanov et al. used it to show the equivalence of adiabatic computation and the quantum circuit model. In this work, we analyze the low energy properties of a class of modified circuit Hamiltonians, which include features like complex weights and branching transitions. For history states with linear clocks and complex weights, we develop a method for modifying the circuit propagation Hamiltonian to implement any desired distribution over the time steps of the circuit in a frustration-free ground state, and show that this can be used to obtain a constant output probability for universal adiabatic computation while retaining the $\Omega(T^{-2})$ scaling of the spectral gap, and without any additional overhead in terms of numbers of qubits. Furthermore, we establish limits on the increase in the ground energy due to input and output penalty terms for modified tridiagonal clocks with non-uniform distributions on the time steps by proving a tight $O(T^{-2})$ upper bound on the product of the spectral gap and ground state overlap with the endpoints of the computation. Using variational techniques which go beyond the $\Omega(T^{-3})$ scaling that follows from the usual geometrical lemma, we prove that the standard Feynman-Kitaev Hamiltonian already saturates this bound. We review the formalism of unitary labeled graphs which replace the usual linear clock by graphs that allow branching and loops, and we extend the $O(T^{-2})$ bound from linear clocks to this more general setting. In order to achieve this, we apply Chebyshev polynomials to generalize an upper bound on the spectral gap in terms of the graph diameter to the context of arbitrary Hermitian matrices.


Introduction
The initial demonstration of QMA-completeness of the local Hamiltonian problem [1] was followed by a period of development during which the main goal was to broaden the class of interaction terms which suffice to make the local Hamiltonian problem QMA-complete [2,3]. These results were motivated in part by a desire to understand the hardness of approximating physical systems that resemble those found in nature, and also by the goal of making the closely related universal adiabatic computation construction [4] more suitable for eventual physical implementation [5]. The success of these efforts have resulted in QMA-complete local Hamiltonian problems with restricted properties such as 2-local interactions [2], low dimensional geometric lattices [3,6], and translational invariance, as well as a complete classification of the complexity of the 2-local Hamiltonian problem for any set of interaction couplings [7].
This great success in classifying the hardness of physically realistic interactions stands in contrast with the relative lack of progress in resolving questions related to the robustness of quantum ground state computation, such as whether fault-tolerant universal adiabatic computing is possible, or to prove or disprove the quantum PCP conjecture [8]. Such questions motivate us to seek (or to limit the possibility of) improvements to the circuit-to-Hamiltonian construction itself, which serves a foundational role in all of the results listed above. Based on ideas by Feynman [9] and cast into its current form by Kitaev [1], the construction remains relatively little changed but has undergone some gradual evolutions throughout its long history [6,10,11,12,13].
To be robust a circuit-to-Hamiltonian construction should not only have a ground space representing valid computations, but intuitively it should penalize invalid computations with as high of an energy as possible. One way of formalizing this condition is to add constraints on the input and the output of the circuit that cannot be simultaneously satisfied under the valid operation of the circuit. If the Hamiltonian enforces the correct operation of the circuit gates, then the input and output constraints that contradict each other should not be satisfiable by any state, and so the ground state energy should increase. This explains why the higher ground state energy associated with non-accepting circuits can be regarded as an energy penalty against invalid computations.
The specific role of the O(T −3 ) scaling of the quantum UNSAT penalty in Kitaev's proof is to show that the local Hamiltonian problem is QMA-hard with a promise gap that scales inverse polynomially in the system size. While there exists a well-defined relation between runtime T and the corresponding Hamiltonian's system size n for any specific set of constructions-e.g. Kitaev's 5-local one-the explicit scaling of this gap with T is not meaningful to the local Hamiltonian problem beyond the fact that it is polynomial, since the local Hamiltonian problem promise gap is parameterized by the number of qubits n, i.e. 1/ poly n. Nevertheless, the scaling of the quantum UNSAT penalty with T is a well defined feature of any particular circuit-to-Hamiltonian construction, and therefore we take the view that it is a reasonable metric for exploring the space of possible improvements to this construction.

Results and Overview
We analyze circuit Hamiltonians with history state ground states consisting of an arbitrary complex superposition of the time steps of the computation, where as usual U T , . . . , U 1 are quantum gates, |ξ is an arbitrary input to the computation, and |ψ is a normalized state, so that in particular π t := ψ * t ψ t is a probability distribution on {0, . . . , T }. Ground states of the form as in eq. (1) arise from modifications to the usual terms of the Feynman circuit Hamiltonian, where |a t |, |b t | ≤ 1 for t = 0, . . . , T . Note that most if not all of the constructions that implement the O(log n)-local interactions with the time register using a k-local Hamiltonian with constant k, such as the domain wall clock which leads to a 5-local circuit Hamiltonian, can be directly applied to the modified form (2). In addition to the part of the Hamiltonian which checks the propagation of the circuit, projectors H in := |0 0| ⊗ Π in and H out := |T T | ⊗ Π out can be added to H circuit to validate specific inputs and the outputs of the computation.
More specifically, the ground space of H circuit + H in will be spanned by computations starting from a valid input computation (i.e. those for which |ξ ∈ ker Π in in eq. (1)), and H out will raise the energy of the state |ψ in (1) when U T · · · U 1 |ξ ∈ ker Π out . The magnitude of this frustration between the incompatible ground spaces of H in + H circuit and H out will depend on the circuit encoded by H FK and the specific in-and output energy penalties, i.e. the acceptance probability of the circuit = min |ξ ,|η = η| U T · · · U 1 |ξ . In the following definition we take the Π in and Π out that are used in the standard construction: Π out := |1 1| measures a single qubit, and Π in constrains a fraction of the input qubits to the |+ state and leaves the rest of the input qubits unconstrained.
For a specific set of in-and output constraints and runtime length T , we want to identify the circuit Hamiltonian best suitable to discriminate accepting and rejecting circuit paths, independent of the particular circuits used. Therefore, we let C( , T ) be the set of circuits of size T for which the maximum acceptance probability is for any state obeying the inand output constraints Π in and Π out , i.e.
This leads to our definition of the quantum UNSAT penalty of a circuit-to-Hamiltonian construction, which captures how well a Hamiltonian as in eq. (2) can enforce the input and output penalties described above for an arbitrary circuit.
Note that the quantum UNSAT penalty has a closely-related quantity, the average energy of any local Hamiltonian constraints QUNSAT ψ = e∈E ψ| h e |ψ /|E| for some set of interactions E = {h 0 , . . . , h |E| } and a specific state ψ, as defined in the context of the detectability lemma [14]. We use the term UNSAT penalty to emphasize that it is the energy difference between accepting and non-accepting computations.
Our first step in analyzing the UNSAT penalty of modified Feynman Hamiltonians is to apply the same argument used in the standard construction (cf. [1, sec. 14.4]) to "undo" the computation and show that H circuit is unitarily equivalent to a Hamiltonian which acts trivially on the computational register.
where the clock Hamiltonian H clock is given by Next we apply the same geometrical lemma used in Kitaev's proof to lower bound the UNSAT penalty of modified Feynman Hamiltonians.
The upper bound follows immediately by an operator inequality, and says that the increase in the ground state energy due to the penalty terms is at most bounded by the case when all of the frustration is in the system's time register. The lower bound states that the UNSAT penalty can be increased either by boosting the spectral gap of the clock Hamiltonian, or by amplifying the overlap of the ground state with the beginning and ending time steps of the computation. To see that the overlap with the endpoints of the computation can be made arbitrarily close to one, we prove that it is in fact possible to construct a clock Hamiltonian with an arbitrary distribution as its ground state.

Lemma 3. For any probability distribution µ with support everywhere on its domain
(4) such that H clock is frustrationfree and has a ground space spanned by states of the form eq. (1) with weights ψ t = √ µ t .
Using lemma 3 we exhibit a modified Hamiltonian for a target ground state distribution with π 0 , π T ≥ 1/4, and show that it has a spectral gap that is Ω(T −2 ) to establish our first main result.

Theorem 1. There is a frustration-free modified Feynman circuit Hamiltonian as in eq.
Finally, we show that the scaling of the UNSAT penalty achieved in theorem 1 is the optimal scaling that can be achieved by applying the lower bound in lemma 2 to modified Feynman Hamiltonians of the form in eq. (2).

Theorem 2. Let |ψ be the ground state of a Hamiltonian H with eigenvalues
The rest of the paper is organized as follows. The proofs of lemma 1 and lemma 3 can be found in section 2.1. The construction used for lemma 3 is given in section 2.2 along with some necessary background on Markov chains that will be used in the proof of theorem 1 in section 2.3. In section 3.1 we develop the quantum-to-classical mapping for arbitrary tridiagonal matrices and use it to prove theorem 2. In section 4 we describe the implications of our work for universal adiabatic computation. Finally, in section 5 we discuss the open problem of further increasing the quantum UNSAT penalty and relate it to some of the longstanding goals in the subject of quantum ground state computation.

Non-uniform circuit histories for improving the UNSAT penalty 2.1 Analysis of modified Feynman Hamiltonians
Proof of lemma 1. Since W is a linear operator the calculations we need to check for lemma 1 are the same as in the standard unweighted case [1, ch. 14.4]. As a reminder, and so the claim of lemma 1 follows by linearity.
Proof of lemma 2. Kitaev's geometrical lemma provides the starting point for the lower bound (5).

Lemma (Kitaev's geometrical lemma). Let A, B ≥ 0 be positive semi-definite operators, both with a zero eigenspace, and such that ker
the minimal non-zero eigenvalue of A and B, respectively. Then where θ is the angle between the kernels of A and B.
For us, A = H in + H out , and B = H circuit , and in this section we use the freedom to shift the energy in eq. (2) to set E(H circuit ) = 0 (since the system can be frustrated this means the local terms may no longer be positive semi-definite, but this will not present a problem in applying the geometrical lemma above because H circuit itself is positive semidefinite). Denote the projector onto the kernel of the penalty terms A with Π pen := We first want to bound the angle θ between the kernels of the propagation and penalty Hamiltonians.
where we have saturated Cauchy-Schwartz in the third line ( * ). To bound the first inner product, we observe that if ψ 2 0 ≥ ψ 2 T , picking |φ ∈ ker Π in gives the bound max where ϑ is the angle between supp Π in and supp U Π out U † . This angle can be lower-bounded by the acceptance probability of the circuit: Similarly, if ψ 2 0 < ψ 2 T , one can show an upper bound of π 0 cos ϑ + π T . We thus obtain an overall upper bound In Kitaev's lemma, we thus obtain a lower bound and the claim follows.

Symmetrized Metropolis Hamiltonians with target GS distributions
In this section we describe our construction which fulfills lemma 3. We review most concepts as needed but assume the reader has some familiarity with Markov chain transition matrices as can be found in any textbook on the subject, such as [15].
Proof of lemma 3. Given a probability distribution π with support everywhere on its domain S = {0, . . . , T } we can construct an irreducible Markov chain P with state space S and by taking the Metropolis chain with transition probabilities given by for all i ∈ S (setting the expressions P 0,−1 and P T,T +1 to zero) and P (i, i ) = 0 for all i, i ∈ S with |i − i | > 1. The transition matrix P := i,i ∈S P i,i |i i| is not a symmetric matrix, but a well established technique in the analysis of Markov chains is to relate P to a symmetric matrix, The two matrices are related by the fact that if v 0 |, · · · , v T | are the left eigenvectors of P with eigenvalues λ 0 = 1 ≥ λ 1 ≥, . . . , ≥ λ T ≥ 0 then |w i := t∈S v i |t t|v 0 −1/2 |t satisfies A|w i = λ i |w i . Therefore A has the same eigenvalues as P , and in particular it has largest eigenvalue 1 corresponding to the eigenvector |w 0 with components satisfying t|w 0 = t|v 0 1/2 = √ π t . Therefore H = I − A is a non-negative Hermitian matrix with ground state that has energy zero and components √ π t in the time register basis, as claimed.
Markov chain spectral gaps. Substantial efforts been devoted to characterizing spectral gaps of Markov chains. A particularly fruitful characterization procedes by defining a quantity called the conductance, which determines the spectral gap within a quadratic factor, The lower bound in eq. (12) is usually called Cheeger's inequality, and it was initially discovered in the analysis of manifolds [16] before being adapted to the setting of Markov chains [17]. In the next section we will use this method to lower bound the spectral gap of the Symmetrized Metropolis Hamiltonian corresponding to a particular non-uniform stationary distribution.

Proof of Theorem 1
Set π 0 = π T = 1/2 and π t = (2T − 2) −1 for t = 1, . . . , T − 1, and define H circuit as the Symmetrized Metropolis Hamiltonian corresponding to this probability distribution. Keeping with tradition [18,4,6], we exhibit H as a T + 1 by T + 1 matrix in the time register basis, A few low energy eigenstates of this Hamiltonian are illustrated in figure 1. Since π 0 and π T are Ω(1) it only remains to check that the spectral gap ∆ H of the clock Hamiltonian is Ω(T −2 ), which since this spectral gap is equal to the spectral gap of the Metropolis transition matrix P that is reversible with respect to π we can apply Cheeger's inequality (12).
The goal is to show that every subset S of S has large conductance, so we divide the proof into cases corresponding to the different possibilities for the subset S. First if S = {0} then since P 0,1 = (8T − 8) −1 so Φ(S) is Ω(T −1 ), with similar statements holding for S = {T } and S = {0, T }. Now if S = {1, . . . , T − 1} is non-empty there must be at least one t ∈ {1, . . . , T − 1} such that there is a t ∈ S c with P t,t ≥ 1/4, and since π t = (2T − 2) −1 this shows that Φ(S) is Ω(T −1 ) in this case as well. Therefore ∆ P is Ω(T −2 ) by (12) and since ∆ H = ∆ P this concludes the proof of theorem 1.

Proof of Theorem 2
The proof of Theorem 2 is based on applying a sharp spectral gap bound for birth-anddeath Markov chains to a quantum-to-classical mapping that has been studied previously in the closely related context of universal adiabatic computation [4] and the complexity of stoquastic Hamiltonians [19,20]. A new feature of our application is the realization that this quantum-to-classical mapping defines a Markov chain even for tridiagonal Hamiltonian matrices with arbitrary complex entries, while previous applications have been restricted to cases for which H has all non-positive off-diagonal matrix entries in the time register basis. In this section we continue with the notation of (2), but now we use the freedom to shift the energy to set a t ≥ 0 for all t, and so the ground state energy E will in general satisfy 0 ≤ E < 1.
In the following lemma we will show that the P t,t are transition probabilities for an irreducible Markov chain on {0, . . . , T }, i.e. in particular that they are all non-negative. First, observe that if H is stoquastic as in previous applications, then G is a non-negative matrix in the time register basis, ψ has non-negative amplitudes in this basis by the Perron-Frobenius theorem and so P t,t is explicitly non-negative. Here we show that even when G contains arbitrary complex matrix entries (while being tridiagonal) we still have P t,t ≥ 0, because of cancellations that occur between the matrix elements of G and the amplitudes of the ground state wave function in the time register basis. Continuing with the same notation used in (4),
Before proving the lemma, note that the conditions may be taken to hold without loss of generality, since ψ 0 = 0 or ψ T = 0 immediately implies Theorem 2, or similarly if b t = 0 for some t then |ψ ⊥ := T t=0 ψ ⊥ t |t defined by satisfies ψ ⊥ |ψ = 0 and H|ψ ⊥ = E|ψ ⊥ , which implies ∆ H = 0 and so again Theorem 2 holds in this case 1 . Now turning to the proof of Lemma 4. From H|ψ = E|ψ we have that Since P t,t = 0 when |t − t | > 1, our goal is to show P t,t+1 > 0 for t ∈ {0, . . . , T − 1}, and P t,t−1 > 0 for t ∈ {1, . . . , T }. The first claim P t,t+1 > 0 will follow by showing that E is minimized when ψ t = 0 and ψ t+1 b t ψ −1 t < 0 for all t = 0, . . . , T − 1. The second claim P t−1,t > 0 is then implied immediately since Rearranging eq. (16) yields ψ 1 b 0 ψ −1 0 = E − a 0 , and since E − a 0 is real the value of E implied by this equation alone is minimized when the LHS is negative. This observation will be taken as the base case for an argument by mathematical induction on the finite set {1, . . . , T − 1}. The inductive hypothesis is that the value of E implied by considering only equations 0 through i in the list eq. (17) is minimized when ψ t b t−1 ψ −1 t−1 is negative for 1, . . . , t, and this will be used to show that the minimum value of E that satisfies equations 0 through t + 1 in eq. (17) will be achieved when ψ t+1 b t ψ −1 t is negative as well. Using the fact that ψ t = 0 from the inductive hypothesis we may express eq. (17) as Since E − a t is real and * is negative by the inductive hypothesis, the value of E implied by equations 0 through t + 1 in the list eq. (17) will indeed be minimized by taking ψ t+1 b t ψ −1 t to be negative. This establishes the inductive claim and completes the proof of Lemma 4.
Having established that P t,t ≥ 0 we now list several standard facts which have been previously applied to P when G is non-negative, which can also be seen in the present case by direct computation: 1. P is a stochastic matrix, i.e. T t =0 P t,t = 1 for all t ∈ {0, . . . , T }, and therefore it can be regarded as the transition matrix of a discrete time Markov chain.
2. The largest eigenvalue of P is equal to 1 and it corresponds to the unique principal eigenvector |π = T t=0 |ψ t | 2 |t . The probability distribution π t := t|π is the stationary distribution of the corresponding Markov chain.
3. The Markov chain defined by P is reversible with respect to its stationary distribution, . Since this is the complete list of eigenvectors of P we have shown that the spectral gaps of H and P satisfy The relation eq. (21) means that we can apply techniques developed for upper bounding the spectral gap of Markov chains to the problem of upper bounding the spectral gap of H. A non-trivial example of such an upper bound is eq. (12): if the overlap of the stationary distribution with the end points |0 and |T is constant then we can immediately see that the conductance Φ is O(T −1 ) by the fact that the stationary distribution is normalized, and this implies that ∆ H is O(T −1 ). It turns out we can obtain an even tighter bound by using a characterization of spectral gaps that applies specifically to birth-and-death chains [21], which we state here as a lemma.
In the present case we are seeking a lower bound on in order to have an upper bound on the gap. To simplify the formulas we assume that the stationary distribution of the weighted history state is symmetric around t = T /2 (otherwise the problem divides into two similar cases). Since we are seeking a lower bound on we can ignore the factor of P k,k+1 ≤ 1 in the denominator, and we are also free to replace the maximization over j with any fixed choice of j.
With these simplifications and the choice of j = 1 eq. (23) becomes Applying the inequality of the arithmetic and geometric means yields, and so is Ω(ψ 2 0 T 2 ), and from eq. (22) we have that the spectral gap ∆ P is O( −1 ) and so ∆ H · ψ 2 0 is O(T −2 ) as claimed. with Ω(1) probability on the endpoints can be used to increase the probability that measuring the time register will collapse the computational register of the system into the final time step of the computation. This provides an alternative to the theoretical solution of "padding the end of the computation with identity gates" that is normally used to raise the probability of sampling from the final time step of the computation. Padding the length of the computation with identity gates is relatively expensive in practical terms when the time register is encoded using local interactions (such as the domain wall clock) because the clock must be represented in unary, meaning the number of clock qubits scales linearly with the total length of the (padded) computation. Achieving an overlap of δ ≈ 1 with the final step of the computation by padding the system with identity gates requires a total of O(T /(1 − δ)) clock qubits, however one can instead prepare the weighted history state with π T = δ using only T clock qubits. The price that one has to pay for this improvement is in an increase in the precision of the couplings needed to implement eq. (2) now must scale like O(T −1 ), as seen in (13). This is a reasonable trade off, however, since the total number of qubits is generally the limiting factor in most experiments.

Relation to universal adiabatic computation
Furthermore, theorem 2 can be interpreted as proving that the standard universal adiabatic construction plus the weighted endpoint modification made above is in a sense optimal for Hamiltonians of the form (2). First, the problem of upper bounding the spectral gap of universal adiabatic constructions was addressed before [22] by combining the quantum lower bound for unstructured search with the technique of spectral gap amplification. This previous work found a generalÕ(T −1 ) bound on the spectral gap of any adiabatic Hamiltonian, ã O(T −2 ) gap for any frustration-free adiabatic Hamiltonian, and finally anÕ(T −2 ) bound on the spectral gap of modified Feynman Hamiltonians of the form (2) when the weights near the endpoints satisfy a reasonable assumption for any adiabatic computation. Our theorem 2 corroborates this last result by showing a tight O(T −2 ) upper bound on the spectral gap and the minimum overlap of the weighted history state with either endpoint of the computation.

Outlook
One of the main aims of the present work is to motivate new ideas in quantum ground state computation by focusing on the quantum UNSAT penalty as a metric for the improvement of circuit Hamiltonians. In this section we discuss a range of open problems related to the UNSAT penalty, in case that they are more tractable or lead to a different perspective on some of the open challenges facing this field. One difficulty is that there is at present no general abstract formulation of what it means for a Hamiltonian to have a ground space of circuit histories, as further improvement could involve alterations to the tridiagonal form of the clock Hamiltonian (one such construction allowing for computational paths that include branching, concurrency, and loops is given in [13]). Therefore we describe these open problems without specifying a precise form for future circuit-to-Hamiltonian constructions e.g. how the number of local terms might scale, and so we are implicitly discussing relative energy penalties that are not simply made larger by e.g. increasing the overall norm of the Hamiltonian.
The classical baseline. The classical Cook-Levin theorem encodes the history of a classical circuit into the satisfying assignment of a 3-SAT formula. If the computation has T time steps, then the associated constraint satisfaction problem has O(T ) local terms and if each has a constant norm than the classical UNSAT penalty is immediately Ω(1). Therefore we ask: is it possible for a circuit Hamiltonian containing O(T ) local terms of bounded norm, which may be of a form more general than (2), to achieve an UNSAT penalty that is independent of the length of the computation?
Macroscopic UNSAT penalty. Building on the previous question which asks whether the UNSAT penalty can be made independent of the length of the computation, we further ask whether the UNSAT penalty can be made to scale macroscopically with the number of qubits n in the computation. Specifically, is there a circuit Hamiltonian with O(poly(n)T ) local terms that achieves a poly(n) UNSAT penalty that is independent of T ? Such a construction could be a useful step towards fault-tolerant adiabatic computation. An intuition for this connection can be gained by considering a construction for energetically encoded faulttolerant classical computation, whereby each logical bit could be encoded as an arrangement of spins in a self-correcting model (e.g. the 2D Ising model), so that the UNSAT penalty could have a macroscopic scaling (i.e. with the number of physical spins representing each logical bit) that is independent of T . It is a testament to Feynman's great legacy that an idea first introduced in 1987 has had such a profound impact for wide scope of research, from condensed matter physics to quantum computation, and that despite the growth of the field of Hamiltonian complexity his original construction continues to remain essentially unchanged to date. We do not know whether or where limitations of improving the circuit-to-Hamiltonian construction will be reached, but hope that our contribution will help to push this boundary a little further.