Fixed-Point Adiabatic Quantum Search

Alexander M. Dalzell, Theodore J. Yoder, and Isaac L. Chuang Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: February 7, 2017) Fixed-point quantum search algorithms succeed at finding one of M target items among N total items even when the run time of the algorithm is longer than necessary. While the famous Grover’s algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property — the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder et al. [1] gave an optimal gate-model search algorithm with the fixed-point property. Meanwhile, it is known [2] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non- fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.

I. INTRODUCTION search algorithm [9, 10] since the success probability remains high even when λ > w (i.e. the algorithm was run The remarkable discovery of Grover’s algorithm [3], longer than necessary). which solves the simplest of search problems quadrati- Fixed-point search algorithms can be used for error- cally faster than a classical computer, has helped fuel correcting schemes [11], oblivious amplitude amplifica- interest in quantum computing for the last two decades. tion [12], or situations where a natural lower bound on λ Grover’s algorithm finds one of M target items among exists, such as preparation of the relatively-prime state N total items in O(pN/M) time with high probability (see section VI). More generally, knowing the existence by repeated application of an operation called the Grover of even one target state provides a lower bound λ ≥ 1/N. iterate on an initial superposition of all of the items. The Additionally, Aaronson and Christiano [8] explain how a quantum state lies in an N dimensional Hilbert space, fixed-point algorithm might be a strategy for counterfeit- but the symmetry of the problem allows for a reduction to ers of quantum money to amplify imperfect copies. two dimensions. In this picture, the Grover iterate can be The O(pN/M) scaling of Grover’s algorithm has been interpreted as a rotation which moves the quantum state shown to be optimal [13], but this speedup is not re- closer to the direction of the target items. However, too stricted to the gate model; a similar speedup is found many applications causes the state to overrotate; knowl- in continuous models. For example, Farhi and Gut- edge of λ = M/N, the fraction of target items, is needed mann introduced the Hamiltonian oracle model [14] and to choose the right number of iterations. showed how an analogue of search displays the quadratic How do we find a target item when λ is unknown, while speedup. Likewise, the speedup is reproduced [2] for maintaining the Grover-like quadratic speedup? One so- searching in the model of Adiabatic Quantum Compu- lution is to first estimate λ using quantum counting al- tation (AQC) [15]. But these algorithms rely on knowl- gorithms [4–6]; another involves choosing the number edge of λ, the first to decide when to terminate the evolu- of iterations randomly from an exponentially increasing tion and the second for defining the adiabatic interpola- arXiv:1609.03603v2 [quant-ph] 4 Feb 2017 range of integers until a target state is found [7]. Al- tion schedule. Since Hamiltonians generate rotations, we though these methods have Grover-like scaling, they re- might expect to be hindered by the same problem of over- quire measuring the system and repreparing the initial rotation that plagues Grover’s algorithm in the absence state. In contrast, Aaronson and Christiano [8] give a so- of knowledge of λ. Indeed, search in the Hamiltonian or- lution which prepares a state, albeit a mixed state, arbi- acle model faces this exact problem. However, for AQC, trarily near to the target state subspace avoiding the need we will show how this issue is avoided by presenting a for measurement and requiring only knowledge of a (typ- fixed-point version of the adiabatic search algorithm with ically small) lower bound w for the fraction λ. Similarly, run time which displays Grover-like scaling (meanwhile, under the same assumption w ≤ λ, Yoder et al. [1] pro- other versions are shown not to share these properties). vide an algorithm which coherently prepares the (pure) In the context of AQC, fixed-point algorithms imply uniform superposition |Ei over the M target states with a robustness to systematic errors in the initial Hamilto- arbitrarily small error: it produces |Ei with fidelity at √ √ nian. Moreover, a continuous algorithm might fit nat- least 1 − δ2 in only O(log(1/δ)/ w) time, thus exhibit- urally into experiment, or perhaps provide a constant- ing the quadratic speedup. The algorithm is a fixed-point factor speedup in implementation over a gate-model Fixed-Point Adiabatic Quantum Search 2

fixed-point algorithm. Indeed, implementing fixed-point point. Exact expressions such as this are a rare occur- search as a continuous algorithm, we gain continuous con- rence among results regarding the adiabatic theorem. trol over the lower bound w, which in the gate-model al- Third, we describe a gate-model algorithm in the same gorithm [1] is related to the number of Grover iterates oracle framework of [1] which simulates the adiabatic and is therefore discrete. search algorithm and we bound its failure probability Rigorously proving that a version of the adiabatic (Theorem 4). This bound implies that the simulation search algorithm is fixed point requires precise state- of the adiabatic search algorithm with standard schedule ments about the adiabatic theorem, which are often elu- retains the fixed-point property and Grover-like scaling, sive. Many AQC works simply take a heuristic approach yielding an alternative gate-model fixed-point algorithm. to the adiabatic theorem [15, 16], and do not explicitly We argue that the existence of such a gate-model simu- evaluate or bound the error in the adiabatic approxi- lation that does not compromise the quantum speedup is mation, instead declaring the approximation valid if the not obvious. Hamiltonian varies slowly in comparison to the square of its eigenvalue gap. Other works, including [2], use ex- II. THE ADIABATIC SEARCH ALGORITHM plicit bounds on the error which have since been shown to be incorrect in some cases [17, 18]. More rigorous approaches have been taken. For example, in [19–24], the In this section, we set up the adiabatic search prob- error is written as a series in powers of 1/T where T is the lem and discuss the adiabatic theorem. Continuous- total run time. Rezakhani et al. [24] do this explicitly for time search takes place in an N-dimensional Hilbert the adiabatic search algorithm, while some of the other space, a superposition of the orthonormal states treatments rely on assumptions or consider cases which |0i , |1i ,... |N − 1i. Some number, M, of these states are not applicable to the adiabatic search algorithm. This have been marked, dubbed target states, and our goal is type of expansion is useful for understanding the asymp- to obtain one of them. To do so, we are given access to totic behavior of the algorithm, but less so for finding (scaled multiples of) the oracle Hamiltonian I − P where explicit bounds on the error when T and λ are finite. In P is the projector onto the space spanned by the M tar- [25] and [26], a bound on the error is proved in terms of get states, and I is the identity operator. We see that the eigenvalue gaps and derivatives of the Hamiltonian, any target state is an eigenstate of I − P with eigenvalue but these works are concerned mostly with establishing 0, while any non-target state is an eigenstate with eigen- a polynomial relationship between T , the gap, and the value 1. This is not the only oracle Hamiltonian which adiabatic error. These bounds are not tight enough to works; any Hamiltonian whose ground state is a solution imply Grover-like scaling for the search algorithm. It to the search problem is sufficient. would be preferable to develop an intuitively-motivated The adiabatic solution to this search problem begins and explicit upper bound on the algorithm’s failure prob- the system in the equal-superposition state ability which is both tight enough to give reliable scaling N−1 1 X estimates and easy to evaluate for finite run times of the |Bi = √ |xi (1) adiabatic search algorithm. N x=0 We develop exactly such a method for analysis of the adiabatic search algorithm motivated by the geometric at time t = 0, and further applies the Hamiltonian intuition that a two-level system evolving by a time- H(t) = (1 − s(t))H + s(t)H (2) dependent Hamiltonian is equivalent to a spin precessing 0 1 about a varying magnetic field. Using this framework we where derive three main results. First, we consider an arbitrary interpolation schedule H0 = I − |Bi hB| (3) and find an explicit expression (Theorem 1) which up- H1 = I − P (4) per bounds the failure probability of the algorithm. We evaluate the upper bound for several different families of and s(t) is a continuous and monotonically nondecreasing schedules, parameterized by w, a lower bound for λ, and function of time such that s(0) = 0 and s(T ) = 1. We see , a slowness parameter. We show examples of schedules that the beginning state |Bi is the ground state of the which have the fixed-point property but lack Grover-like initial Hamiltonian H0. We are assuming, as is standard scaling, and vice-versa. The “standard” schedule consid- [14, 15], that the ability to apply certain Hamiltonians ered by [2] is shown to have both Grover-like scaling and individually also implies the ability to apply their sum. the fixed-point property simultaneously (Theorem 3). Experimentally this is quite justifiable (see e.g. [27, 28]). With this physical setup, the system, represented by Second, we modify our argument to produce an ex- |ψ(t)i evolves based on the Schrödingerequation, which act expression for the failure probability for a particular is given by schedule (Theorem 2), which reproduces the quantum speedup for search – in fact it is even faster than the d i |ψ(t)i = H(t) |ψ(t)i . (5) standard schedule by a constant factor – but is not fixed dt Fixed-Point Adiabatic Quantum Search 3

The adiabatic theorem suggests that if the Hamiltonian λ = M/N be the fraction of target states allowing us √ √ is changed slowly enough, the system will remain approx- to write |Bi = 1 − λ E¯ + λ |Ei. The projector P imately in the instantaneous ground state of the Hamil- acts on this subspace like |Ei hE|. As the state evolves tonian. If this is true, at time t = T , the system will by (5) it will always remain in this subspace. We change be approximately in the ground state of H1, and when basis by letting |0i = cos(µ) E¯ + sin(µ) |Ei and |1i = √ measured will yield a target state with high probability. − sin(µ) E¯ +cos(µ) |Ei, with cos(2µ) = 1 − λ. In this basis Eqs. (3) and (4) can be written as A. The Adiabatic Theorem 1 1 H = I − nˆ · ~σ (8) 0 2 2 0 The adiabatic theorem is used to approximate the evo- 1 1 H = I − nˆ · ~σ (9) lution of a quantum state subject to a continuously vary- 1 2 2 1 ing Hamiltonian. Suppose a system represented by |ψ(t)i is exposed to a time-dependent Hamiltonian H(t), as is where true for the algorithm we discuss in this paper. Further, √ √ nˆ0 = λ xˆ + 1 − λ zˆ (10) suppose H(t) has spectral decomposition √ √ nˆ1 = λ xˆ − 1 − λ zˆ (11) H(t) |φn(t)i = En(t) |φn(t)i (6) are normalized vectors and ~σ is the vector of Pauli ma- where En(t) and |φn(t)i are the instantaneous eigenen- trices (X,Y,Z). The purpose of the basis change is to ergies and eigenstates of the Hamiltonian, with E0 < maken ˆ0 andn ˆ1 symmetric about the xy-plane. E1 . . . < EN . The adiabatic theorem states that if the From Eq. (2), we write the Hamiltonian in this 2D state begins in the ground state |ψ(0)i = |φ0(0)i, then space as it will stay approximately in the ground state (up to a global phase) throughout the evolution |ψ(t)i ≈ |φ (t)i, 1 ∆λ 0 H(s) = I − nˆ · ~σ (12) provided that the Hamiltonian is changing sufficiently 2 2 slowly. The criteria for slowness has been a notorious point of confusion for AQC. The general heuristic for where adiabaticity used in early papers on AQC, such as [15], ∆ (s) = p1 − 4s(1 − s)(1 − λ) (13) relates the rate of change of the Hamiltonian to the en- λ 1 √ √ ergy gap ∆(t) ≡ E1(t) − E0(t) between the ground state nˆ(s) = λ xˆ + 1 − λ(1 − 2s)z ˆ (14) and the first excited state ∆λ = sin(θ(s))x ˆ + cos(θ(s))ˆz (15) 2 |hφ1(t)| H˙ (t) |φ0(t)i| ∆(t) . (7) and This makes the notion of slowness less vague, but still √ tells us nothing about how exactly the slowness of the (1 − 2s) 1 − λ θ(s) = arccos(ˆn · zˆ) = arccos (16) algorithm affects the error in the approximation. More- ∆λ over, a closer examination reveals this criteria is not even completely correct [17, 18]. However, more rigor- is the angle the normalized vectorn ˆ makes with the z- ous treatments of the adiabatic theorem have been done axis. Using Eqs. (13) and (16), it can be verified that [19–21, 23–25]. In this paper, we shall use the heuristic p condition to inform our choice of interpolation schedule dθ 2 λ(1 − λ) = 2 . (17) s(t), but we do not rely on it mathematically. By analyz- ds ∆λ ing the algorithm geometrically, we find a rigorous upper bound on the error of the approximation in terms of the We will also find it useful to describe states by their schedule s(t). The requirement that this bound on the position on the Bloch sphere. A normalized vectorr ˆ can error be small can be used as a more rigorous and less be written as vague replacement for Eq. (7). rˆ = sin(ξ) cos(φ)ˆx + sin(ξ) sin(φ)ˆy + cos(ξ)ˆz (18)

The state which is mapped to the pointr ˆ on the Bloch B. Geometry of the Algorithm sphere is

As in the gate-model analysis of Grover’s algorithm, |rî = cos(ξ/2) |0i + eiφ sin(ξ/2) |1i (19) the symmetry of the adiabatic search algorithm allows us to reduce the state space to two dimensions, spanned It is quick to show that by |Ei and E¯ where |Ei is the equal superposition over the M target states (or “end” state), and E¯ is the equal rˆ · ~σ |rî = |rî (20) superposition over the N − M non-target states. We let rˆ · ~σ |−rî = − |−rî (21) Fixed-Point Adiabatic Quantum Search 4

Thus, referring to Eq. (12), the eigenvectors of H(s) are the probability of measuring a target state at the end of |nˆ(s)i and |−nˆ(s)i, and the gap between their eigenvalues the algorithm as a function of λ is ∆λ. 2 By viewing the state of the system |ψ(t)i by its po- P (λ) = |hE|ψ(T )i| . (22) sition on the Bloch sphere and the Hamiltonian H(t) = We also define the error amplitude δ(λ) so that P (λ) = I/2 − ~v(t) · ~σ/2 by the “Hamiltonian vector” ~v(t), it is 1 − δ(λ)2. apparent how we can think the system as a spin-1/2 par- We note that when λ = 0, there are no target states, ticle exposed to a magnetic field of strength ∆λ pointing and when λ = 1, all the states are target states, so re- in then ˆ direction. The tip of the magnetic field vector gardless of schedule, P (0) = 0 and P (1) = 1. ∆λnˆ associated with the Hamiltonian traces out a verti- In the next section, we discuss how the error amplitude cal line in the xz-plane as shown in Figure 1. With this δ(λ) depends on the choice of schedule s(t) and whether in mind, we can see intuitively how the algorithm works. the algorithm has the fixed-point property. The spin of the particle precesses on the Bloch sphere around the magnetic field at a rate proportional to the field strength. At first, the spin and field are aligned in III. SCHEDULING AND THE FIXED-POINT the direction of |Bi. The field is rotated from the begin PROPERTY state |Bi to the end state |Ei. The precession of the spin around the field causes the field to drag the spin along A. Fixed-point property with it as it changes, so at time T the spin is approximately in the same direction as the field, which corre- For our analysis, we consider families of schedules sponds to the desired final state |Ei. This is a specific s(t; , w) parameterized by and w. The parameter example of the adiabatic theorem in two-dimensions. is typically small and represents how fast the interpolation occurs. The w parameter represents a lower bound on the fraction of target states λ. Each schedule family does not depend on λ since we assume in general that we do not know λ precisely, all we know is the lower bound w ≤ λ. This is a reasonable scenario: if we know that at least one item is marked, then λ ≥ 1/N. Moreover, some problems admit tighter lower bounds, such as the problem of preparing the relatively-prime state, which we discuss in section VI. Together, and w determine the total run time T by requiring s(0; , w) = 0 and s(T ; , w) = 1. In section III C, we give three examples of different families of schedules. We denote the family of success probability functions which correspond to the family of schedules s(t; , w) by P (λ; , w) and, by extension, the family of error ampli- tudes by δ(λ; , w) = p1 − P (λ; , w). We now formally stipulate conditions on P (λ; , w) that make the search al- FIG. 1: Geometric picture of the algorithm. This is the gorithm under schedule s(t; , w) a fixed-point algorithm. xz-plane of the Bloch sphere for λ = 1/4. The Hamiltonian vector ∆ nˆ associated with H(s) follows λ Fixed-point property. The adiabatic search algo- the dotted line between the begin state and the end rithm operating under schedule family s(t; , w) has the state. fixed-point property if there exists a function f() independent of λ and w such that P (λ; , w) ≥ 1 − (f())2, The gap ∆λ(s) can be interpreted as the length of the for all λ ≥ w and f() → 0 as → 0. Hamiltonian vector as it changes linearly from |Bi to |Ei √in Figure 1. Notably, ∆λ(0) = ∆λ(1) = 1 and ∆λ(1/2) = If we have an adiabatic fixed-point search algorithm, λ. and we wish to make P (λ) ≥ 1 − δ2 for all λ ≥ w, we The only piece left to the algorithm is what to choose need only make small enough so that f() ≤ δ. Thus for the function s(t). Regardless of the form of s(t), the we view and w as independent input parameters which Hamiltonian vector will follow the linear trajectory from control two features of our success probability function the begin state to the end state; the function s(t) deter- P (λ). The parameter controls an upper bound on the mines how fast it moves along this trajectory. We call failure probability f()2, and w controls the range over s(t) the schedule of the algorithm. which this success guarantee is valid. The price paid for Once the schedule has been specified, the final state a decrease in w or is an increase in the run time T . |ψ(T )i can be computed by evolution via the Schrödinger Although the notion of a schedule and the slowness equation (5). For a given schedule s(t), we can compute parameter might not make sense in the gate model, the Fixed-Point Adiabatic Quantum Search 5 adiabatic fixed-point definition is designed to mirror the Thus, the requirement that δ(λ) be small combined with definition in the gate model. A gate-model fixed-point Theorem 1 serves as an alternative adiabatic condition algorithm, such as that presented in [1], is constructed for s(t) which is more concrete (and more rigorous) than given parameters δ and w to succeed with probability Eq. (24). P (λ) ≥ 1 − δ2 so long as λ ≥ w. We are also interested in the scaling of the algorithm. Theorem 1. If the adiabatic search algorithm is run The algorithm√ has Grover-like scaling if the run time T = according to schedule s(t) then δ(λ) ≤ d0 + d1 where O(1/ w). p It is possible for a family of schedules to lack the fixed- d0 = 2 λ(1 − λ)s ˙(0) (25) point property if δ(λ; , w) cannot be bounded by a func- ! Z T d pλ(1 − λ) ds tion only of . It is also possible for a family of sched- d = dt (26) 1 3 ules to possess the fixed-point property but lack Grover- 0 dt ∆λ dt like scaling√ if the dependence of T on w is worse than O(1/ w). We present examples of these cases in section III C, where we consider three different families of Theorem 1 is proved in section IV A. The error is re- schedules. Determining if a certain schedule family has lated to the derivatives of the schedule; we think of d0 as the fixed-point property requires analysis of the error in the initial error due to the choice of derivative at t = 0 the adiabatic approximation. The tool we use for this and d1 as the additional error accrued during the algo- purpose is Theorem 1, presented in the next section. rithm. If we require the bound in Theorem 1 on δ(λ) to be small, then d0 and d1 must be small. Since d0 is small, the quantitys ˙pλ(1 − λ)/∆3 must be small for s = 0 and B. General bound on error probability λ since d1 is also small, that quantity must remain small for all s, implying Eq. (24) is satisfied. The heuristic condition for validity of the adiabatic However, the converse relationship is not true. Eq. (24) theorem in Eq. (7) is inadequate for several reasons. implies that d0 is small but does not necessarily imply First, we have not rigorously defended it, and it has ac- that d1 is small. It is possible thats ˙(t) is small through- tually been shown not to be a sufficient condition for out the algorithm whiles ¨(t) is not small. Thus, the con- adiabaticity in all cases [17, 18]. But also, it fails to pro- dition d0 + d1 1 is strictly stronger than the heuristic vide information about the error amplitude δ(λ), besides adiabatic condition derived from Eq. (7). that it is small. In particular, it does not allow us to understand how our choice of schedule s(t) affects the error amplitude δ(λ). Such information would be practically C. Schedules useful for actually running the adiabatic search algorithm and also allows us to understand the asymptotic behavior In this section, we apply Theorem 1 to three families of of δ(λ) as T → ∞ and as λ → 0. schedules. Our first two families fail to have both Grover- We wish to provide this information by finding an up- like scaling and the fixed-point property, although we per bound for δ(λ) in terms of the schedule s(t). As a note how they can be made to have one of the two traits. starting point, we can evaluate Eq. (7) using the geomet- The third family avoids the issues facing the first two and ric framework we set up in section II B. From Eq. (12), has both properties. we see that the ground and first-excited states of H(s) are |nî and |−nî, so the left-hand-side of Eq. (7) is 1 1. Constant-speed schedule |h−nˆ| H˙ |nî| =s ˙(t)|h−nˆ| (ˆn − nˆ ) · ~σ |nî| 2 0 1 √ First, we consider the constant-speed schedule family =s ˙(t) 1 − λ|h−nˆ| Z |nî| s (t; , w) defined by ds /dt = . This definition does p c c c c λ(1 − λ) not depend on the w parameter. Boundary conditions =s ˙(t) . (23) ∆λ imply Tc = 1/c, and where our computation relies on Eqs. (2), (8-11),√ and sc(t; c, w) = ct = t/Tc. (27) (19). We note that since the minimum of ∆λ is λ, this quantity is at mosts ˙(t). Since the eigenvalue gap is ∆λ, We can evaluate the bound in Theorem 1√ exactly, finding that d = 2pλ(1 − λ) and d = 2 1 − λ(λ−1 − Eq. (7) reads √ 0 c 1 √ λ)c. So our bound reads δ(λ; c, w) ≤ 2c 1 − λ/λ. 3 ds ∆λ Letting δ be the maximum error amplitude for λ ≥ w, our p . (24) dt λ(1 − λ) bound says that δ = maxλ≥w δ(λ; c, w) < 2c/w, which diverges as w approaches 0 with c constant. Hence, Now, in Theorem 1, we present an upper bound on the there is no function f(c) such that our bound guarantees error amplitude δ(λ) which can be computed given s(t). δ ≤ f(c); this suggests the algorithm is not fixed point. Fixed-Point Adiabatic Quantum Search 6

Even equipped with a tighter upper bound than the one Theorem 1 to get an upper bound on the failure proba- we present, it cannot be possible to bound δ ≤ f(c) with bility. Supposing that λ = w, we calculate d0 = 2f and limc→0 f(c) = 0: the error amplitude δ(λ; c, w) has d1 = 0, meaning no w-dependence, so limw→0 δ ≥ limλ→0 δ(λ; c, w) = 1. Thus, the constant-speed schedule does not have the δ(w; f , w) ≤ 2f . (31) fixed-point property. However, we succeed in getting a concrete upper bound When λ = w, the error probability is bounded by a function of f , and the run time Tf has Grover-like scaling in on the error probability in terms of both w and c, if w! To show that the algorithm is fixed point, we would not c alone. Decreasing w with constant c causes the bound on the error probability to rise, but this effect need this to also be true when λ > w. In fact, for the fast schedule, we can modify the ar- can be canceled by a roughly proportional decrease in c. The fixed-point definition is not satisfied since both pa- guments for the bound in Theorem 1 to give an exact rameters must be adjusted to keep the error probability expression for the error amplitude when λ = w: low, but we can use this fact to create a related family of schedules which is fixed point. Theorem 2. If the adiabatic search algorithm is run 0 0 0 according to the fast schedule given by (30), then Consider the schedule sc(t; c, w) defined by dsc/dt = 0 cw. The relationship between the primed and un- q  0 1 + 42 primed schedules is simply c = cw, so our bound 2f f 0 δ(w; , w) = sin φ (32) states that δ = maxλ≥w δ(λ) < 2c. For this schedule, f q  w 0 0 1 + 42 2f Tc = 1/(cw) = O(1/(δw)). It is fixed point but it lacks f Grover-like scaling. where

r1 − w 2. Fast schedule φ = arctan . (33) w w Our first attempt failed to achieve a quantum speedup. In our second attempt, we make the schedule faster when the gap is large, and slower when the gap is small. The Theorem 2 is proved in section IV B. This result is idea of varying the speed according to the gap was first significant. If the value of λ is known, we can run the proposed in [2], but our schedule varies in proportion to adiabatic search algorithm with the fast schedule setting the cube of the gap, while that in [2] varies in proportion parameter w = λ, and we have an exact analytic expres- to the square (and is presented as our third example). sion for the error in the algorithm implying that it has Additionally, previous works have not considered these Grover-like scaling in the fraction λ. schedules in the context of the fixed-point property and Varying the speed of the interpolation to be faster the lower bound w ≤ λ. when the gap is larger allows us to recover the Grover The fast schedule family sf (t; f , w) is defined by speedup seen in the circuit model. But what about when λ is unknown? Suppose all 3 dsf ∆w we know is that λ ≥ w. The heuristic condition from = f . (28) dt pw(1 − w) Eq. (24) now reads pw(1 − w) ∆3 This schedule is designed to make the left and right sides λ (34) f p 3 of Eq. (24) proportional by the constant f for all sf when λ(1 − λ) ∆w λ = w. From this equation and the boundary conditions sf (0) = 0, sf (Tf ) = 1, it can be shown that the total This is problematic when λ is large, and sf is close to 0 time must be or 1. For example, fixing s√f = 0 (so ∆λ = ∆w = 1), and √ λ = 1/2, we have √f w. Looking at Eq. (29), we 1 − w see that if f = O( w) then Tf = O(1/w), meaning the Tf = √ (29) f w quadratic speedup is lost. We confirm this notion with the more con- and the schedule is crete bound in Theorem 1. We evaluate p p s d0 = 2f λ(1 − λ)/ w(1 − w) and d1 = 1 1 2t w 3/2 −3/2 p p s (t; , w) = − 1 − . 2f (1 − w λ ) λ(1 − λ)/ w(1 − w). In this f f 2t 2 2 2 Tf 1 − (1 − ) (1 − w) p p Tf case, δ(λ; f , w) ≤ d0 + d1 ≤√4f λ(1 − λ)/ w(1 − w) √ (30) which increases like O(f / w) as w → 0. We are Since Tf = O(1/ w), this schedule has Grover-like scal- unable to bound the error probability by a function of f ing, a quadratic speedup over the classical search com- alone, suggesting the algorithm is not fixed point. Other plexity. However, the Grover-like scaling is only useful if techniques could, in principle, yield a tighter bound the algorithm actually succeeds. We apply the result of on the error probability, but numerical simulations of Fixed-Point Adiabatic Quantum Search 7

the algorithm confirm that the algorithm lacks the �� fixed-point property, so any such attempt would suggest Fast the same result. �� As in the constant-speed case, the schedule can be Standard modified to become fixed point. Defining s0 (t; 0 , w) by Constant-speed f f �� 0 0 3 the relation dsf /dt = f ∆w leads to the relationship T 0 = 1/(0 w) and the bound δ(λ; 0 , w) ≤ 20 when- � � f f f f �� (�) ever λ ≥ w, meaning the primed algorithm is fixed point. � � In this case, the cost of the fixed-point property is the � 0 �� loss of the Grover-like scaling, since Tf = O(1/w). � � �� 3. Standard schedule �� /� The third schedule family we present does not have FIG. 2: A comparison of schedules s(t) at w = 1/20 this problem. The standard schedule family s (t; , w), s s plotted with respect to normalized time t/T . Both the is defined by the relationship fast and standard schedules speed up when the gap is large and slow down when it is small, with the fast dss 2 = s∆ (35) schedule exhibiting this behavior more strongly. As dt w w → 0, the fast and standard schedules approach the Using the boundary conditions ss(0) = 0, ss(Ts) = 1, it discontinuous function that is 1/2 for t/T ∈ (0, 1) and 0 can be shown that (1) for t/T = 0 (1). As w → 1 both schedules approach the constant-speed schedule. The inset shows the total φw time T taken (in units of 1/) by these search algorithms Ts = p (36) s w(1 − w) as a function of w. The fast algorithm is the fastest.

where φw is given by Eq. (33), and the schedule is r as a heuristic condition for adiabaticity or something 1 1 w 2t similar. Some go further and prove asymptotic state- ss(t; s, w) = − tan (1 − )φw (37) 2 2 1 − w Ts ments about the relationship between the error probability and the run time [19, 25], and some even compute By comparison with Eq. (29), we note that the fast sched- an asymptotic 1/T expansion of the error probability ule is a factor of π/2 faster than the standard schedule [20, 21, 23, 24], but often this series is truncated after for small w and s = f . The three schedules are plotted a finite number of terms (which is valid only for large in Figure 2. The standard schedule is so-named because T ). Here we give an explicit upper bound on the error it is the schedule first introduced in [2]. It also appears for general schedule in Theorem 1, and apply this bound as the “constant-norm interpolation” in [24]. to the standard schedule in Theorem 3, which confirms We first examine the heuristic condition from Eq. (24), asymptotic notions, but is also valid for finite values of 3 2 p which reads s ∆λ/(∆w λ(1 − λ)), and is satisfied the run time T and parameter w. for all λ ≥ w so long as s is small. This suggests that Theorem 3 rigorously establishes that the standard the error will be small for all λ√≥ w while the run time schedule is a fixed-point algorithm with Grover-like scal- complexity is Grover-like O(1/ w). ing. Since δ = O(s), Eq. (36) implies that To be more rigorous we apply the bound in Theorem 1. The computation is not as straightforward as for the φw 1 Ts = = O √ (39) other two schedules but is carried out in Appendix A and O(δ)pw(1 − w) δ w summarized by Theorem 3. Comparing this adiabatic algorithm to the fixed-point√ Theorem 3. If the adiabatic search algorithm is run algorithm in [1], which has scaling O(log(2/δ)/ w), we with parameters w and s according to the standard can see that it has similar scaling in w, but exponentially schedule given by Eq. (37), then worse scaling in δ. However, if the goal is simply to find the target state, δ = max δ(λ; s, w) ≤ 2s (38) logarithmic scaling in δ can be achieved by repetition of w≤λ≤1 our algorithm. To see this, suppose we fix δ2 = 1/3, and repeat the algorithm log(δ02)/ log(1/3) times. The algorithm fails only if each individual run fails, which Theorem 3 is unique among previous rigorous treat- occurs with probability δ02. The total time of this algo- ments of the adiabatic theorem, as it relates to AQC. As rithm is logarithmic in the eventual error amplitude δ0: 0 0 mentioned, some previous papers [15] merely use Eq. (7) Ts = O(log(1/δ )). Fixed-Point Adiabatic Quantum Search 8

Additionally, other schedules are possible which might where θ is given by Eq. (16). At any instant of time, the have better scaling in δ. In particular, it is discussed relationship between |ψi and |φi is simply a rotation of in [24] how setting the first k time derivatives of s(t) to the Bloch sphere by angle −θ about they ˆ-axis. zero at the beginning and end of the algorithm makes the first k terms in the 1/T expansion of the error amplitude Lemma 1. |φ(t)i evolves according to the Schrödinger vanish, suggesting that δ = O(1/T k+1). If all of the equation time derivatives vanish at s = 0 and s = 1, we might even be able to reproduce the logarithmic dependence d i |φ(t)i = Hφ |φ(t)i (41) on δ without resorting to repetition. These notions are dt explored in an asymptotic sense in [24], but no bound for where H is the effective Hamiltonian given by finite T and w is given as we do in Theorem 1. φ 1 m H = I − nˆ · ~σ (42) φ 2 2 φ with q IV. GEOMETRIC INTUITION AND PROOFS 2 ˙ 2 m = ∆λ + (θ) (43)

and Theorems 1 and 2 are proved in this section, while The- orems 3 and 4 are proved in the appendix. Throughout, 1 nˆ = (θ˙yˆ + ∆ zˆ) (44) when we refer to a Hamiltonian H = I/2 − ~v · ~σ/2 in a φ m λ geometric context, we mean the Hamiltonian vector ~v.

Geometric Intuition. At its core, the method for arriving at Theorems 1 and 2 is very intuitive. We view the Proof of Lemma 1. This lemma follows from appli- Hamiltonian and state vectors geometrically on the Bloch cation of the Schrödingerequation (5) on the definition sphere. As the Hamiltonian is changed, the state vector of |φi in Eq. (40). strays away from the Hamiltonian vector. We would like d θ˙ θ θ to find how far away it moves, which we quantify by com- i |φ(t)i = − Y + exp(i Y )H exp(−i Y ) |φ(t)i puting an upper bound on the angle between the state dt 2 2 2 and Hamiltonian vectors at the end of the algorithm. We do this by looking at the state from the point of view Using the geometric expression (12) for H, we have of the Hamiltonian; we change coordinates so that the θ θ Hamiltonian always points in thez ˆ direction, and the exp(i Y )H exp(−i Y ) 2 2 state is nearly in thez ˆ direction. This coordinate trans- 1 ∆ θ θ formation is time-dependent, which gives rise to another = I − λ exp(i Y )(ˆn · σ) exp(−i Y ) term in the effective Hamiltonian. For the fast schedule, 2 2 2 2 1 ∆ the direction of the effective Hamiltonian remains fixed, = I − λ Z although its magnitude changes. Since the state vector 2 2 in these coordinates precesses about the effective Hamil- and thus, tonian, the angle between the state vector and effective Hamiltonian is constant in time, allowing for exact calcu- 1 ∆ θ˙ H = I − λ Z − Y lation of the error amplitude δ(w), and leading to Theo- φ 2 2 2 rem 2. For general schedule, the direction of the effective Hamiltonian is not constant in time, but it changes very and the lemma is proved. little. As a result, the angle between the state vector and the Hamiltonian changes very little and can be bounded, The term ∆λZ/2 in Hφ comes from rotation of the yielding Theorem 1. Hamiltonian back to thez ˆ-axis. The term θY/˙ 2 comes from the time-dependence of this rotation. The vital mathematical concept is the time-dependent change of coordinates which rotates the Hamiltonian A. Proof of Theorem 1 back to thez ˆ direction, and is described by Lemma 1. The change of coordinates is defined by the state To prove Theorem 1, we will make two coordinate changes and use Lemma 1 twice. The effective Hamil- θ tonian vectors in each frame are depicted graphically in |φ(t)i = exp(i Y ) |ψ(t)i (40) 2 Figure 3. Building off of Lemma 1, we define Fixed-Point Adiabatic Quantum Search 9

Proof of Lemma 2. Applying the left-hand-side of Eq. (48) to Eq. (50) gives

r i r eiq −q˙ cos − r˙ sin |0i + 2 2 2 (a) (b) (c) r i r r ei(q+p) −q˙ sin + r˙ cos − p˙ sin |1i 2 2 2 2 FIG. 3: Geometric guide to the proof of Theorem 1. We perform two time-dependent coordinate changes. and applying the right-hand-side gives (a) The state |ψi evolves by Hamiltonian H, consistent 1 r r r with Figure 1. (b) The state |φi is related to |ψi by eiq cos − m cos + eipχ˙ sin |0i + Eq. (40) and evolves by Hamiltonian Hφ. (c) The state 2 2 2 2 |ξi is related to |φi by Eq. (47) and evolves by 1 r r r eiq eip sin +χ ˙ cos + eipm sin |1i . Hamiltonian Hξ. Each Hamiltonian H = I/2 − ~v · ~σ/2 is 2 2 2 2 represented by the vector ~v in the ﬁgure. These quantities must be equal. We can immediately cancel the eiq factor. Now, the imaginary parts of the coeﬃcient of |0i for both the left and right-hand-sides must be equal, giving the equation: ˙ χ(t) = arctan(θ(t)/∆λ(t)) (45) 1 r 1 r − r˙ sin = χ˙ sin(p) sin to be the angle betweenn ˆφ andz ˆ, allowing us to rewrite 2 2 2 2 (44) as r˙ = −χ˙ sin(p)

nˆφ = sin(χ)ˆy + cos(χ)ˆz, (46) |r˙| ≤ |χ˙| (51) noting the similarity to Eq. (15). If the interpolation is which proves the lemma. ˙ slow, θ, and hence χ will be small, son ˆφ will be close to zˆ. Physically, Lemma 2 is telling us that the angle r Now, we repeat the process again, this time usingn ˆφ between |φi and the effective Hamiltonian Hφ cannot in place ofn ˆ and rotating in the yz-plane instead of the change faster than the angle χ between the effective xz-plane. We let Hamiltonian Hφ andz ˆ. This fact could have been proved χ using a purely geometric argument. |ξ(t)i = exp(−i X) |φ(t)i (47) The success probability (22) can be expressed in our 2 new coordinates using the relationships (40) and (47). and by invoking the result from Lemma 1 (except in the yz-plane instead of xy-plane), we see that |ξ(t)i evolves P (λ) = |h0|φ(T )i|2 = |hξ(0)|ξ(T )i|2 (52) according to We define d i |ξ(t)i = Hξ |ξ(t)i (48) r(0) + r(T ) dt A = (53) 2 with 1 1 1 m χ˙ Using Eq. (50), we can see that Hξ = I − nˆξ · ~σ = I − Z + X (49) 2 2 2 2 2 P (λ) = |hξ(0)|ξ(T )i|2 ≥ cos2(A) ≥ 1 − A2 (54) We note that |ξ(0)i = cos(χ(0)/2) |0i − i sin(χ(0)/2) |1i. The most general possible equation for |ξ(t)i is R T Since 0 rdt˙ = r(T ) − r(0), we have r(t) r(t) |ξ(t)i = eiq(t) cos |0i + eip(t) sin |1i . 1 Z T 2 2 A = r(0) + rdt˙ 2 (50) 0 1 Z T Note that r is the angle betweenz ˆ and |ξi on the Bloch ≤ χ(0) + |χ˙|dt. (55) sphere, but since |ξi and |φi are related by the rotation 2 0 (47), which preserves angles, r is also the angle between the direction of the effective Hamiltonian Hφ and the Plugging Eq. (17) into the definition of χ in Eq. (45), we p 3 state |φi. see that χ(t) = arctan(2 λ(1 − λ)s ˙(t)/∆λ(t) ). How- By plugging (50) into the Schrödingerequation (48), ever, when u > 0, arctan(u) ≤ u and |d(arctan(u))/dt| = 2 R we can prove the following: |u/˙ (1 + u )| ≤ |u˙|. Thus, χ(0) ≤ d0 and |χ˙|/2 ≤ d1 where d0 and d1 are given by Eqs. (25) and (26). Since Lemma 2. |r˙| ≤ |χ˙|. δ(λ) = p1 − P (λ) ≤ A, this proves the theorem. Fixed-Point Adiabatic Quantum Search 10

B. Proof of Theorem 2 unlikely that this latter sort of evolution could be exhibited by any adiabatic algorithm, but perhaps the former Theorem 2 gives an exact expression for the error prob- could be exhibited by a gate-model algorithm. Fortu- ability where Theorem 1 only gives an upper bound. The nately, simulating the adiabatic search algorithm in the structure of the proof is the same as that for Theorem 1. gate-model is simply done using a Trotter formula, as we We define χ(t) as in Eq. (45) for the fast schedule with explain in this section. We adapt our geometric analysis λ = w, but as can be verified by Eqs. (17) and (28), to show that the resulting gate-model algorithm can be made to have the fixed-point property, and thus act as χ = 2f is constant throughout time. Son ˆφ = sin(χ)ˆy + cos(χ)ˆz is fixed. In this frame, the effective Hamiltonian an alternative to the algorithm in [1] with evolution more does not change direction, though its strength varies and closely resembling that of the adiabatic search algorithm 2 ˙ 2 1/2 2 1/2 itself. is given by m(t) = (∆ + (θ) ) = ∆w(1 + 4 ) . w f We will do this simulation in two steps: first, we will Thus, the angle between the state vector andn ˆφ should remain constant throughout the evolution. Mathemat- discretize the continuous Hamiltonian H(t), so it is con- ically, this follows from Lemma 2, sinceχ ˙ = 0 implies stant over small time intervals δt, then we will simulate angle r is constant. It is this fact that allows for exact the discrete Hamiltonian using a Trotter formula. Sup- treatment of the error probability. This is not a coin- pose we divide the total time T into l intervals of width δt = T/l. Then we let t = j δt, and s = s(t ) for some cidence. The proportionality between the gap ∆w and j j j the angular velocity θ˙ of H(t) is intimately related to schedule s(t). The discretized Hamiltonian is the adiabatic condition in Eq. (7) and the definition of the fast schedule in Eq. (28). We define |ξi as in Eq. (47) Hd(t) = H(δtbt/δtc). (58) but since χ is constant the coordinate change is no longer Thus, H agrees with H whenever t = t for some j, and time-dependent. We see that H = I/2 − m(t)Z/2. d j ξ H is constant between t and t for all j. At t = 0 we have |ξ(0)i = cos(χ/2) |0i − i sin(χ/2) |1i. d j j+1 For a given interval j, the effect of applying the Hamil- Evolution by Hξ(t) simply applies a relative phase on the tonian Hd is the unitary operator |1i component of |ξi, so up to a global phase, |ξ(Tf )i = T cos(χ/2) |0i − i exp(−i R f m(t)dt) sin(χ/2) |1i. (j) 0 Ud = exp(−iH(tj) δt) (59) The phase given by the integral can be evaluated as = exp − i (1 − sj) H0 δt − i sj H1 δt . (60) Z Tf Z 1 1 q 2 p dsf m(t)dt = 1 + 4f w(1 − w) 2 0 f 0 ∆w If δt is small, then we can use a Trotter formula to approximate this unitary as a product of two unitaries: q 2 φw = 1 + 4f (56) f (j) Ut = exp(−i (1 − sj) H0 δt) exp(−i sj H1 δt). (61) where, incidentally, the integral is the same one used to (j) (j) 2 arrive at Eq. (36), with φw given by Eq. (33). Finally, It can be shown that Ud (t) = Ut (t) + O(δt ) [29]. we can evaluate the success probability P (w; f , w) = The benefit of Ut is that it is a product of two unitaries 2 |hξ(0)|ξ(Tf )i| . We let the integral in Eq. (56) be denoted which are easy to implement in a circuit. Exponentiating by p and evaluate H0 is equivalent to a rotation about the begin state in the Bloch sphere framework we introduced above. Like- 2 2 2 P (w; f , w) = |cos (χ/2) + exp(−ip) sin (χ/2)| wise, exponentiating H1 is a rotation about the end state. Specifically, we have: = 1 − sin2(χ) sin2(p/2) (57) (j) U = SB(αj)SE(βj) (62) thus δ(w; f , w) = sin(χ)|sin(p/2)| where sin(χ) = t 2 /(1 + 42 )1/2, so the theorem has been proved. f f up to an overall phase, where

−iα SB(α) = I − (1 − e ) |Bi hB| V. SIMULATION IN THE GATE MODEL = exp(−iα) exp(iαH0) (63) +iβ What is the relationship between the standard- SE(β) = I − (1 − e ) |Ei hE| schedule adiabatic fixed-point algorithm and the gate- = exp(iβ) exp(−iβH1) (64) model fixed-point algorithms from [1]? They bear little resemblance beyond the fixed-point property. The evolu- and αj = −(1 − sj)δt and βj = sjδt. The circuit that (j) tion of the state in the adiabatic case is smooth and stays implements Ut is the same one discussed in [1] and is nearly in the xz-plane of the Bloch sphere throughout the shown in Figure 4. It makes use of the discrete oracle evolution, while the gate-model algorithm subjects the U, which flips an ancilla bit when fed a target state: state to discrete rotations through large angles causing U |xi |0i = |xi |1i if |xi is a target state, and otherwise it to jump around all parts of the Bloch sphere. It seems acts as the identity. As an example, suppose we are using Fixed-Point Adiabatic Quantum Search 11

2 the standard schedule ss(t; s, w) given by Eq. (37), then, algorithm√ with standard schedule, P (λ) ≥ 1−δ with δ = 2 recalling that we have split Ts into l equal time steps of 3.1 δt + 2s(1 + δt /25). The number of oracle queries size δt, we can say that associated with this sequence of length l is L = 2l+1 and l = bT/δtc. Thus, we can express the query complexity δt δtr w 2j α = − − tan (1 − )φ (65) in terms of the desired values for w and δ: j 2 2 1 − w l w r 2φ 1 δt δt w 2j L = 1 + 2bT /δtc = 1 + w = O √ . βj = − tan (1 − )φw (66) s p 3 2 2 1 − w l δt w(1 − w) δ w (68) Thus, Theorem 4 shows that the simulated adiabatic search algorithm with standard schedule is a fixed-point algorithm which shows Grover’s quadratic speedup. This gives an alternative sequence of α and β angles (see Eqs. (65) and (66)) that can be used in place of the fixed- point sequence given in [1]. It is known that adiabatic computation can be simulated with polynomial-overhead [30], but this algorithm shows that the adiabatic search algorithm can be simulated while maintaining the same scaling in w. FIG. 4: The circuit which simulates the discretized We briefly mention an overview of how Theorem 4 is Hamiltonian Hd over one time interval of width δt, proved. One approach would be to bound the error intro- (j) performing the unitary operation Ut . The gate Zζ duced by discretization (the difference between applying applies the operation exp(−iζZ/2). the true Hamiltonian H and the discrete Hamiltonian Hd), and then bound the error induced by Trotterization (j) (j) Now we can formally define the gate-model algorithm: (the difference between applying Ud and Ut ). We did not find this approach successful for the following reason: suppose the algorithm is run for time T with l time steps, Simulated Adiabatic Search Algorithm. 2 Inputs: schedule s(t) (where s(0) = 0, s(T ) = 1), δt so δt = T/l. The Trotterization error is O(δt ) per time step, making the total error O(lδt2) = O(T δt). Since Procedure: Prepare the initial state |ψ0i to be |Bi. √ √ (j) T = O(1/ λ) and the error must be O(1), δt = O( λ), Apply circuit U to |ψ i to yield |ψ i for j = 0, 1, 2, t j j+1 and thus the number of time steps l = O(1/λ), which ..., l − 1, where l = bT/δtc. Measure the system in the does not have Grover-like scaling. The discretization er- computational basis. ror is not problematic in this way. Output: The post-measurement state is a target state 2 Instead, our proof finds an alternate continuous Hamil- with probability P (λ) ≡ |hψl|Ei| , where λ is the fraction of target states. tonian Ht which, upon discretization, generates the ex- (j) act evolution operations Ut applied in the simulated adiabatic search algorithm and given by Eq. (61). This We claim that the simulated adiabatic search algo- Hamiltonian Ht, which is approximately equal to H, fol- rithm, using the standard schedule ss(t; s, w) given by lows a slightly modified but still continuous path from Eq. (37), is also fixed point with Grover-like scaling. We the begin state |Bi to the end state |Ei. In particular, prove this by bounding the error in the simulated adia- this path leaves the xz-plane. We use the same strategy batic search algorithm for general schedule s(t) and then as in the proof from Theorem 1 to bound the error in- evaluating this bound for the standard schedule (recall duced by the continuous algorithm defined by application δ(λ) = p1 − P (λ)). of Ht. This error does have Grover-like scaling. Then, we discretize Ht, giving Htd, and bound the error induced Theorem 4. If the simulated adiabatic search algorithm by discretization. is run according to schedule s(t) and step size δt, then While the first approach views Trotterization as an √ error-inducing step and tries to find an upper bound on 2 δ(λ) < 3.1 δt + (d0 + d1)(1 + δt /25) (67) this error, the second approach capitalizes on the fact that the Trotterization error at each time step is cor- where d0 and d1 are functions of λ given by Eqs. (25) and related in a way that can be interpreted as simply an (26) from Theorem 1. alternate path from |Bi to |Ei on the Bloch sphere and hence should also be a valid adiabatic algorithm. As Theorem 4 is proved in Appendix B. As δt approaches T gets longer with constant δt, the first approach finds 0, this bound approaches the bound in Theorem 1. If that the error accrues while the second approach does s(t) is the standard schedule (37), then Theorem 3 says not. The alternate path is a function only of δt and not that d0 + d1 ≤ 2s, so for the simulated adiabatic search of the schedule s(t) (which defines how fast the path is Fixed-Point Adiabatic Quantum Search 12 traversed) and hence a longer run time does not result in (or total run time in the adiabatic model). This is a more error due to Trotterization under this approach. quadratic speedup over the algorithm in [32] which does not make use of a quantum search algorithm. On an adiabatic computer, this would entail running the adiabatic VI. APPLICATIONS OF THE FIXED-POINT search algorithm using the standard schedule. Moreover, PROPERTY the fixed-point algorithm allows the state |Φi to be pre- pared to arbitrary fidelity without the need for measure- In this section, we elaborate on two direct applications ment. Crucially, we only need to know the lower bound of fixed-point algorithms: preparation of the relatively- in Eq. (71) for λ = φ(J)/J for the algorithm to work. prime state and oblivious amplitude amplification. Other applications are mentioned briefly in the introduction. B. Fixed-point oblivious amplitude amplification

A. Preparation of the relatively-prime state Oblivious amplitude amplification [12] is a technique for implementing, on a state |ψi, a desired unitary V For a specific example of a problem which might benefit that we cannot construct directly with a larger unitary U from a fixed-point algorithm, we look to number theory. that we can construct, at least when given some ancilla Given an integer J, we wish to prepare the relatively- qubits. The “oblivious” in the name comes from the prime state, the superposition over states which are rel- fact that amplitude amplification still works despite not atively prime to J. having the full ability to reflect about the initial state |ψi. Thus, this procedure is particularly useful when 1 X |Φi = |xi (69) the initial state |ψi is not only unknown but also not pφ(J) gcd(x,J)=1 renewable — we have only one copy and cannot or would prefer not to make another. For instance, this is the case where φ is the Euler totient function. That is, φ(J) is in Hamiltonian simulation [12]. the number of integers relatively prime to J. This can be For convenience, we consider |ψi to be an n-qubit state done by beginning in the equal superposition |Bi, run- and the extension to Hilbert space to be m-qubits. Then, ning the classical Euclidean algorithm (which is efficient) write the start state as [12] for calculation of the greatest common divisor (gcd) while √ √ ⊗m ⊗m ⊥ in superposition, and copying the result to an ancilla reg- U |0i |ψi = λ |0i V |ψi + 1 − λ Φ . (72) ister ⊗m J−1 The target state is |Φi = |0i V |ψi. We note that for X |Bi |0i → |xi |gcd(x, J)i (70) oblivious amplitude amplification to work, it is required of U that λ in Eq. (72) be independent of |ψi. Given this x=0 setup, the question is, how many uses of U do we need Then, measurement of the ancilla register yields outcome to make |Φi with probability 1 − δ2? 1 and the state |Φi with probability φ(J)/J. The process The naive classical attempt would measure the first is repeated until a 1 outcome is obtained. The algorithm m-qubits, succeeding with probability λ in getting |Φi. is efficient because there is a lower bound [31] on φ(J)/J However, upon failure the state will generally be de- stroyed, and given only one copy of |ψi, we will not be φ(J) 1 able to retry except perhaps in special cases when |ψi > (71) γ 3 can be recovered from the remains after measurement. J e log log J + log log J In [12] Berry et al. give an exact quantum algorithm for where γ ≈ 0.577 is the Euler-Mascheroni constant. so oblivious amplitude amplification√ in the case of known λ the state |Φi will be produced with high probability after using the expected number O(1/ λ) of uses of U. They only O(log log J) repetitions of the Euclidean algorithm. argue that, if Π = (|0i h0|)⊗m⊗I, the operations UeiπΠU † We can reformulate the algorithm as a search algo- and eiπΠ act as the reflection about the start and reflec- rithm, with λ = φ(J)/J and the Euclidean algorithm tion about the target states, respectively. Thus, Grover playing the role of the oracle. In the gate model, the ora- iterates constructed from these reflections act as expected ⊥ cle consists of doing the euclidean algorithm and flipping on the 2-dimensional span of |Φi and Φ . an ancilla bit if the gcd is 1. In the adiabatic model, However, it is not immediately clear how to extend the oracle is a Hamiltonian HJ such that HJ |xi = 0 if oblivious amplitude amplification to when λ is unknown. gcd(x, J) = 1 and HJ |xi = |xi otherwise. Thus, the Standard Grover algorithms in the unknown-λ setting ground state of HJ is |Φi. use a set of sequences of Grover iterates that increase ex- Running a fixed-point search algorithm with lower ponentially in length, with a measurement after each to bound parameter w given by Eq. (71) will prepare the determine whether the target state has been found [5]. superposition√ of target states |Φi with high fidelity using Each such measurement (now performed on the m-qubit only O( log log J) queries to the Euclidean algorithm ancilla) will, like in the classical approach, destroy the Fixed-Point Adiabatic Quantum Search 13 state if it fails. Moreover, at least one measurement is lation of a fixed-point adiabatic algorithm with Grover- likely to fail during the process of ramping up the se- like scaling can yield a fixed-point gate-model algorithm quence lengths. which retains the quantum speedup and acts as an alter- A workable approach is to conduct a coherent version native to the algorithms in [1, 8]. of this exponential step-up in sequence length. For the Fixed-point search algorithms provide fault-tolerance problem of standard Grover search, this is essentially the to certain systematic errors and misestimations of λ, and idea from the last section of [33], where amplitude es- also find applications in certain problems with natural timation is used. With great enough precision (a lower lower bounds, like preparation of the relatively-prime bound w ≤ λ is required here to ensure enough preci- state, and even in counterfeiting quantum money [8]. sion), the estimation will, with high probability, yield a That fixed-point Grover-like algorithms exist in the state that has overlap 1/2 with the target state. Exact adiabatic model extends these benefits to AQC but also quantum search can from there complete the algorithm. gives us continuous control over the parameters of the Applying this approach to oblivious amplitude amplifica- algorithm w and which we did not have in the gate- tion, we get an algorithm√ that succeeds with probability model. This continuous control might be useful to create 1−δ2 using O(1/(δ w)) applications of U. Yet, not only a scheme which estimates the value of λ by binary search- is the scaling with δ poor, but there is also not a clear ing the interval [0, 1] using the adiabatic search algorithm adiabatic version of this approach. with different parameters, or different schedule families However, fixed-point search can also solve the prob- altogether. Such a scheme would provide a novel way lem of oblivious amplitude amplification with unknown to perform quantum counting [4], but also, potentially, a λ (and a known lower bound w ≤ λ). If the opti- practical method for calibrating adiabatic quantum com- mal fixed-point√ algorithm [1] is used this will take just puters. O(log(1/δ)/ w) uses of U. To see that this works is Direct applications aside, the method by which we to realize that Ue−iαΠU † and eiβΠ implement partial re- proved our results is an equally important takeaway from flections about the start and target states, respectively, this work. The technique was motivated by the intuition following exactly the same reasoning as in [12]. Since that a two-dimensional system exposed to a Hamiltonian fixed-point search in the gate-model is just unitary ap- is equivalent to a spin immersed in a magnetic field and plication of these partial reflections, it will produce the can be treated geometrically on the Bloch sphere. In this same amplification profile as a function of λ as it does in picture, the state vector precesses around the Hamilto- the non-oblivious case. As an aside, a procedure using nian vector, and by making a series of time-dependent the digital-filter engineering that is also behind the fixed- coordinate changes we demonstrated an upper bound on point search [1] has recently improved the Hamiltonian the angle between the state and Hamiltonian vectors at simulation from [12] by bypassing oblivious amplitude the end of the algorithm. For the fast schedule when amplification entirely [34]. λ = w, this method actually allowed for the Schrödinger An adiabatic oblivious search algorithm is also just as evolution to be solved exactly. straightforward as the non-oblivious version, except with This approach contrasts with other rigorous methods a modified Hamiltonian. One should construct H = (1 − of dealing with the adiabatic approximation, such as the † s(t))HB + s(t)HE, where now HE = Π and HB = UΠU asymptotic expansion of the error. Our bound is explicit (so that e−iHB = Ue−iΠU †). Assuming one can do this, and useful even in situations where the parameters are the adiabatic search schedules s(t) we have presented can finite, giving us more than just asymptotic information be applied to the oblivious case without change. The about the algorithm. The ability to compute the error modified HB may even be easier to implement than a exactly (Theorem 2), albeit in a special case, is a unique full reflection about the initial state as you would find in result for AQC. a non-oblivious adiabatic search, as it can limit coupling Further work might explore other families of schedules between the n-qubit and m-qubit subsystems to just the to achieve better dependence of T on the error ampli- bare essential coupling represented by U. tude δ, modify our geometric method to also find a lower bound on the error in the adiabatic approximation, or extend our ideas to bound the error in the adiabatic ap- VII. CONCLUSION proximation in more than two dimensions.

Inspired by the fixed-point algorithm in [1], we won- dered whether the search algorithm in the adiabatic Acknowledgments model could be made to have the fixed-point property. We found that the landscape of search algorithms in the We thank Guang Hao Low for interesting discussions adiabatic model is similar to that in the circuit model: and perspective especially concerning Hamiltonian simu- some algorithms are fixed point but lack Grover-like scal- lation. A.M.D. acknowledges the Paul E. Gray UROP ing, some have Grover-like scaling but are not fixed point, fund. T.J.Y. thanks the NDSEG fellowship program. and some have both properties. We have given examples The authors also acknowledge funding by the NSF CUA, for each of these cases. Moreover, we showed that simu- and NSF RQCC Project No. 1111337. Fixed-Point Adiabatic Quantum Search 14

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Appendix A: Proof of Theorem 3 We can see this immediately in Case I, because λ ≥ w implies c1/2 ≤ s so d0 + d1 ≤ 2s. Similarly for Case Theorem 3 follows from application of Theorem 1 to III, since c0 ≤ s/2 for all λ we know that d0 + d1 ≤ 4c ≤ 2 . Case II is difficult to analyze analytically, the standard schedule ss(t; s, w) given by Eq. (37). 0 s 2 but the fact that d0 + d1 ≤ 2s for all w and all λ ∈ Sinces ˙s(t) = s∆w, and ∆w(0) = 1, we evaluate d0 [3w(2 + w), (1 + 2w)/3] can be verified graphically. This and d1 from Theorem 1 as proves the theorem. p d0 = 2s λ(1 − λ) (A1) T p ! Z s d ∆2 λ(1 − λ) d = dt s w (A2) Appendix B: Proof of Theorem 4 1 3 0 dt ∆λ ! Z 1 d ∆2 pλ(1 − λ) Since the maximum error amplitude δ is at most 1, the = ds s w (A3) s ds ∆3 theorem is trivially true when δt ≥ 2π. Thus we assume 0 s λ δt < 2π and treat δt as a fixed input parameter. We define Ht(s) as the solution to the equation In the integral for d1, we note that if not for the absolute value, we would be integrating a derivative and the exp(−i(1 − s)H δt) exp(−isH δt) = exp(−iH δt) (B1) answer would simply be the net change in the value of 0 1 t c(s ) := ∆2 pλ(1 − λ)/∆3 from s = 0 to s = 1 (note s s w λ s s To solve this equation, first we assume that H has the that 2c = tan(χ) with χ from (45)). The absolute value t form bars mean that we must count both positive and negative changes as positive, so we can evaluate the integral by ob- 1 γ Ht = I − nˆt · ~σ (B2) serving intervals of ss over which c(ss) is monotonic, and 2 2 summing the absolute value of the change of c(ss) over each of these regions. Now we analyze over what regions wheren ˆt is a unit vector. this c(ss) is monotonic. Then, we write H0 and H1 using Eqs. (8) and (9), First, let expand both sides as a linear combination of {I,X,Y,Z} using exp(iθuˆ · ~σ) = cos(θ)I + i sin(θ)û · ~σ (whereu ˆ is a c = pλ(1 − λ) (A4) unit vector) and equate coefficients, arriving at 0 s √ sw 1 − λ 2 c1/2 = (A5) γ = arccos (cos(a − b) − 2λ sin(a) sin(b)) (B3) √λ δt 3/2 2s λ(1 − w) γδt −1 √ ccrit = p (A6) nˆ · xˆ = sin( ) λ sin(a + b) (B4) 3 3(1 − λ)(λ − w) t 2 s γδt −1 p 1 1 2λ − 3w + λw nˆt · yˆ = sin( ) −2 λ(1 − λ) sin(a) sin(b) (B5) scrit = − (A7) 2 2 2 (1 − λ)(1 − w) γδt √ nˆ · zˆ = sin( )−1 1 − λ sin(a − b) (B6) t 2 Note that d0 = 2c0. There are 3 cases. (1) Case I : λ ≤ 3w/(2 + w). There is one local where a = (1 − s)δt/2 and b = sδt/2. extremum: c(ss) increases from c0 at ss = 0 As expected, in the limit as δt approaches 0, we can to c1/2 at ss = 1/2 and back to c0 at ss = 1. see thatn ˆt approachesn ˆ and γ approaches ∆λ. For finite d0 + d1 = 2c1/2 δt,n ˆ andn ˆt take approximately identical though slightly different paths. In particular,n ˆ stays in the xz-plane for (2) Case II : 3w/(2 + w) < λ < (1 + 2w)/3. There all s, whilen ˆt has a small component in they ˆ direction. are three local extrema: c(ss) increases from c0 We also discretize Ht to form at ss = 0 to ccrit at ss = scrit, decreases to c1/2 at ss = 1/2, then increases back to ccrit at Htd(t) = Ht(bt/δtcδt), (B7) ss = 1 − scrit and finally decreases to c0 at ss = 1. d0 + d1 = 4ccrit − 2c1/2 and we can see that the unitary evolution operator cor- responding to application of Htd between times tj and (3) Case III : λ ≥ (1 + 2w)/3. There is one lo- (j) tj+1 is precisely Ut from Eq. (61). cal extremum: c(ss) decreases from c0 to c1/2 Furthermore, we let |ψt(t)i and |ψtd(t)i be states which at ss = 1/2 then increases back to c0 at ss = 1. evolve by Hamiltonians Ht and Htd, respectively. So, d0 + d1 = 4c0 − 2c1/2 |ψtd(T )i = |ψli, the final state of the simulated adiabatic Thus we have explicitly calculated d0 + d1 as a piecewise search algorithm in the gate model, and hence P (λ) = 2 function of λ (supposing constant w and s). We would |hψtd(T )|Ei| . We bound this probability by first showing 2 like to show that in each of the three cases, we have that |hψt(T )|Ei| is close to 1, and then bounding the d0 + d1 ≤ 2s. distance between |ψt(T )i and |ψtd(T )i. Fixed-Point Adiabatic Quantum Search 16 √ 2 Claim 1. |hψt(T )|Ei| ≥ 1 − ( 2.6δt + (d0 + d1)(1 + the maximum norm of O |αi over all normalized states δt2/25))2 if λ ≥ w. |αi.

2 d Claim 2. |hψtd(T )|ψt(T )i| ≥ 1 − 2δt Lemma 5. |φt(t)i evolves according to i dt |φt(t)i = Hφt |φt(t)i where 2 2 2 2 Lemma 3. If |hQ|Si| ≥ 1 − e1, and |hR|Si| ≥ 1 − e2, ˙ 2 2 I γ θ then |hQ|Ri| ≥ 1 − (e1 + e2) for any normalized |Qi, Hφt = − Z − Y + E (B12) |Ri, |Si. 2 2 2 R T and 0 kEkdt ≤ 1.3δt. We delay the√ proof√ of Lemma 3 until the end of the section. Since 2+ 2.6 < 3.1, Claims 1 and 2 combined Since γ is approximately ∆λ, Hφt is approximately Hφ with Lemma 3 are suﬃcient to prove Theorem 4. Now from Eq. (42) plus a bounded error term E. We write we explain why each claim is true. Hφt = Hφt0 + E. If |φt0i evolves according to Hφt0, we wish to quantify how far away |φt0i is from |φti at time T , which is facilitated by Lemma 6. Proof of Claim 1. Claim 1 is an upper bound on the error resulting from running the continuous search Lemma 6. If |ΨA(t)i evolves by Hamiltonian HA, while algorithm with Hamiltonian Ht instead of H. The claim |ΨB(t)i evolves by Hamiltonian HB, and |ΨA(0)i = highly resembles Theorem 1, and a very similar technique R T |ΨB(0)i then |hΨA(T )|ΨB(T )i| ≥ 1 − kHA − HBkdt. is used in the following proof. 0 As in Theorem 1, we make a time-dependent change of coordinates via a time-dependent unitary transformation Lemmas 5 and 6 tell us that so that the Hamiltonian vector is rotated back to thez ˆ 2 axis. In this case,n ˆt has components in all 3 directions, |hφt0(T )|φt(T )i| ≥ 1 − 2.6δt (B13) so constructing this unitary is more complicated. We express this transformation as a product of 3 unitaries. 2 2 2 Lemma 7. |hφt0(T )|0i| ≥ 1 − ((d0 + d1)(1 + δt /25)) if λ ≥ w, where d0 and d1 are given by Eqs. (25) and (26) |φt(t)i = U3(t)U2(t)U1(t) |ψt(t)i (B8) in Theorem 1.

η1 η2 with U1 = exp(i 2 nˆ · ~σ), U2 = exp(−i 2 (ˆn × yˆ) · ~σ), and θ Compare this result to the conclusion of Theorem 1, U3 = exp(i Y ). U1 is a rotation aboutn ˆ by an angle η1, 2 2 2 that |hφ(T )|0i| ≥ 1 − (d0 + d1) . We have now bounded U2 is a rotation about the vector perpendicular ton ˆ in in Eq. (B13) how close |φt0(T )i and |φt(T )i are, and in the xz-plane by an angle of η2. We will choose η1 and η2 Lemma 7, how close |φt0(T )i and |0i are. We can again so these two operations alone rotate the vectorn ˆt onto use Lemma 3 to conclude that the vectorn ˆ. Then, U3 is the same unitary we used to √ 2 2 2 prove Theorem 1 that rotatesn ˆ back toz ˆ. We find the |hφt(T )|0i| ≥ 1−( 2.6δt+(d0+d1)(1+δt /25)) (B14) rotation angles we need are whenever λ ≥ w. This proves Claim 1. We have shown that if the state is exposed to Ht, instead of H, we still (ˆnt · xˆ) cos(θ) − (ˆnt · zˆ) sin(θ) η1 = arctan (B9) have a search algorithm with bounded error probability nˆt · yˆ that is related only to δt and the schedule s(t). If we relax η2 = arccos ((ˆnt · xˆ) sin(θ) + (ˆnt · zˆ) cos(θ)) (B10) the fixed-point definition to allow algorithms whose error can be bounded by functions of both and δt (but not w), where the components ofn ˆt are given by√ eqs: (B4-B6). then applying Ht(s(t; , w)) would be a fixed-point algo- Recall from Eq. (16)√ that cos(θ) = (1 − 2s) 1 − λ/∆λ = rithm for any schedule for which applying H(s(t; , w)) is nˆ · zˆ and sin(θ) = λ/∆λ =n ˆ · xˆ. Lemma 4 verifies that a fixed-point algorithm. Claim 2 illustrates how the ad- this transformation works. ditional error incurred by using the discrete Hamiltonian Htd instead of Ht can be bounded as well. Lemma 4. I γ (U U U )H (U U U )† = − Z (B11) Proof of Claim 2. First we examine the definition 3 2 1 t 3 2 1 2 2 of Ht to bound kdHt/dsk. We differentiate both sides of Eq. (B1) and cancel factors arriving at

dHt Moreover, when s = 1 we have γ = 1, |nˆi = |Ei, and = −H + e−i(1−s)H0δtH ei(1−s)H0δt ds 0 1 η2 = 0, so up to a global phase U2U1 |Ei = |Ei and −i(1−s)H0δt i(1−s)H0δt 2 = −nˆ0 · ~σ/2 + e (ˆn1 · ~σ)e /2 U3U2U1 |Ei = |0i. Hence, we can rewrite |hψt(T )|Ei| = 2 |h0|φt(T )i| . Now we calculate the eﬀective Hamiltonian dHt ≤ knˆ0 · ~σk/2 + knˆ1 · ~σk/2 = 1 (B15) by which the state |φt(t)i evolves. Note that kOk denotes ds Fixed-Point Adiabatic Quantum Search 17

We define δHt = Ht − Htd. δHt(tj) = 0 for all j, and So, expanded this way, the first rotation is about the y- if tj ≤ t < tj+1, then axis. If the Hamiltonian starts in directionn ˆt, it will finish in direction d d kδHt(t)k = (t − tj) kδHt(c)k ≤ δt kδHt(c)k (B16) dt dt rˆt = ((ˆnt · xˆ) cos(θ) − (ˆnt · zˆ) sin(θ))x ˆ + (ˆnt · yˆ)y ˆ for some c with tj < c < t. But + ((ˆnt · xˆ) sin(θ) + (ˆnt · zˆ) cos(θ))z ˆ d ds d ds d ds kδHt(c)k = kδHt(c)k ≤ k δHt(c)k ≤ Now, if we rotater ˆ about the z-axis by an angle of η = dt dt ds dt ds dt t 1 (B17) arctan((ˆrt · x)/(ˆrt · y)), we will preserve the z-component So, now we have ofr ˆt while rotating the xy-part onto the y-axis. Then, we rotate about the x-axis an angle η1 = arccos(ˆrt · zˆ) Z T Z T ds which will rotate it back to the z-axis. Since the norm kδHtkdt ≤ δt dt = δt (B18) is preserved by all of these rotations, the final result is 0 0 dt I/2 − γZ/2 as claimed. Again invoking Lemma 6, we arrive immediately at the Claim. Proof of Lemma 5. If we plug Eq. (B8) into the Schrödingerequation id |ψti /dt = Ht |ψti, we find that Proof of Lemma 3. Since all states are qubits, this |φti evolves according to Hamiltonian follows from the triangle inequality over a sphere. If the dU3 † dU2 † † dU1 † † † angle between states |Qi and |Ri is 2νqr, between |Qi and H = i U + iU U U + iU U U U U φt dt 3 3 dt 2 3 3 2 dt 1 2 3 |Si is 2νqs, and between |Ri and |Si is 2νrs, then we can 2 2 2 2 † † † say that |hQ|Ri| = cos (νqr), |hQ|Si| = cos (νqs) and +U3U2U1HtU1 U2 U3 2 2 |hR|Si| = cos (νrs). So e1 = sin(νqs) and e2 = sin(νrs). The triangle inequality says that νab ≤ νac + νbc, and The Y term in Eq. (B12) comes immediately from the therefore first term above: ˙ dU3 † θ sin(νqr) ≤ sin(νqs + νrs) i U = − Y dt 3 2 ≤ sin(νqs) cos(νrs) + sin(νrs) cos(νqs)

≤ sin(νqs) + sin(νrs) The I and Z terms will come from the conjugation of Ht as shown in Lemma 4. So we have ≤ e1 + e2 2 2 2 |hQ|Ri| = 1 − sin (ν ) ≥ 1 − (e + e ) dU2 † † dU1 † † † qr 1 2 E = iU U U + iU U U U U 3 dt 2 3 3 2 dt 1 2 3 and the Lemma is proved. dU2 dU1 kEk ≤ + dt dt

Proof of Lemma 4. We will rely on the identity and U1 = cos(η1/2)I + i sin(η1/2)ˆn · ~σ, so dU1/dt = ˙ −η˙1 sin (η1/2) I/2 + i (η ˙1 cos(η1/2)n/ ˆ 2 + sin (η1/2) nˆ) · ~σ exp(−i α rˆ · ~σ)(~v · ~σ) exp(i α rˆ · ~σ) = Rrˆ(2α)~v · σ (B19) whose norm is given by where Rrˆ(2α) represents the operator which rotates vec- dU1 tors aboutr ˆ through an angle of 2α. Ht undergoes three dt subsequent rotations. However, it will be helpful to re- r η˙2 η η˙2 η η arrange the rotations as follows: = 1 sin2 1 + 1 cos2 1 + |nˆ˙ |2 sin2 1 4 2 4 2 2 † † U3U2U1 = (U3U2U3 )(U3U1U3 )U3 η˙1 ˙ η1 1 ˙ ≤ + nˆ sin ≤ (|η˙1| + |η1nˆ|) † η2 † 2 2 2 U U U = exp −i U ((ˆn × yˆ) · ~σ)U 3 2 3 2 3 3 η ˙ = exp −i 2 X Where the second line follows sincen ˆ is orthogonal to nˆ. 2 Likewise, † η1 † U3U1U3 = exp −i U3(ˆn · ~σ)U3 2 dU2 1 η2 d 1 ˙ η ≤ |η˙2|+ (ˆn × yˆ) = (|η˙2|+|η2nˆ|) (B20) = exp −i 1 Z dt 2 2 dt 2 2 η η θ with the ﬁnal equality holding sincen ˆ, nˆ˙ , andy ˆ are mu- U U U = exp −i 2 X exp −i 1 Z exp i Y ∴ 3 2 1 2 2 2 tually orthogonal. Fixed-Point Adiabatic Quantum Search 18

Graphically, we can verify (can be justiﬁed analyti- Also, since d(arctan(u))/dt =u/ ˙ (1 + u2) ≤ u˙, we can say cally), that η1 is monotonically decreasing as a function that of s with η1(s = 0) = η1,max and η1(s) = −η1(1 − s) for all s. Meanwhile, η2 monotonically increases from ! θ˙(0) 1 Z T d θ˙(t) η2(s = 0) = 0 to its maximum η2(s = 1/2) = η2,max, and At ≤ + dt (B24) then returns to η2(s = 1) = 0, maintaining the symmetry γ(0) 2 0 dt γ(t) η2(s) = η2(1 − s) for all s. We not the following, where the subscript max denotes a quantities maximum value with 0 ≤ s ≤ 1. Z T kEkdt 0 ! ! θ˙ θ˙ ∆ 1 Z T 1 Z T = λ ≤ (|η˙ | + |η˙ |)dt + |nˆ˙ |(|η | + |η |)dt γ ∆ γ 2 1 2 2 1 2 λ 0 0 ! ! ! Z T θ˙ θ˙ ∆ θ˙ ∆ η1,max + η2,max ˙ ∂ = ∂ λ + ∂ λ ≤ η1,max + η2,max + |nˆ|dt t γ t ∆ γ ∆ t γ 2 0 λ λ ! ! ≤ (1 + π/2)(η1,max + η2,max) (B21) θ˙ θ˙ ∆ λ ∂t ≤ ∂t R T ˙ γ ∆λ γ max since 0 |nˆ|dt is the arclength of the path traversed by nˆ, which is always less than π. Now, we can verify that ˙ ! θ ∆λ η1,max = η1(s = 0) ≤ δt/4 and η2,max = η2(s = 1/2) ≤ + ∂t ∆λ γ δt/4 as long as δt ≤ 2π (analytically or graphically). max R T Thus, 0 kEkdt ≤ (1 + π/2)(δt/2) ≤ 1.3δt, which proves the Lemma. which allows us to rewrite

! ! θ˙(0) 1 Z T θ˙(t) ∆ Proof of Lemma 6. We let C = hΨ0(T )|Ψ1(T )i, and λ At ≤ + ∂t dt we differentiate both sides ∆λ(0) 2 0 ∆λ(t) γ max dC ˙ ! Z T = −i hΨ (T )| H |Ψ (T )i + i hΨ (T )| H |Ψ (T )i 1 θ(t) d ∆λ(t) dt 0 1 1 0 0 1 + dt (B25) 2 ∆λ(t) 0 dt γ(t) max dC = |hΨ0(T )| H1 − H0 |Ψ1(T )i| ≤ kH1 − H0k dt If we look at ∆λ(s)/γ(s), we can see that it is always at and the lemma follows immediately. least 1, increasing to a maximum at s = 1/2, then decreasing back to 1. If (∆λ(s)/γ(s))max = 1 + x. Graph- ically, we can see that x ≤ δt2/50 whenever δt ≤ 2π Proof of Lemma 7. We go through the same process (can be justified analytically). Moreover, Theorem 1 used to prove Theorem 1, except most of the work has tells us that the first term is at most d0 + d1 given by already been done for us. We define Eqs. (25) and (26). Also, for any quantity u, we have R ˙ ! umax ≤ u(0) + |u|, so (θ(t)/∆λ(t))max ≤ d0 + d1 as well. θ˙(t) χ (t) = arctan (B22) Therefore, we have t γ(t) to mirror Eq. (45). And, as in Theorem 1, we can make At ≤ (d0 + d1)(1 + x) + (d0 + d1)x 2 2 the following bound |hφt0|0i| ≥ 1 − ((d0 + d1)(1 + 2x)) (B26)

|h0|φ i|2 ≥ cos2(A ) ≥ 1 − A2 t0 t t 2 Using δt /50 for x gives the Lemma. 1 Z T At = χt(0) + |χ˙ t|dt (B23) 2 0