Topological invariants for adiabatic quantum computations (Extended abstract)

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1 Topological invariants for adiabatic quantum computations (Extended abstract) CMU quantum group March 21, 2018 Abstract We import the tools of the Morse-Conley (M-C) theory to study adiabatic evolution, the core mechanism in adiabatic quantum computations (AQC). AQC is computationally equivalent to the (pre-eminent paradigm) of the Gate model but less error-prone, and so is ideally suitable to practically tackle a large number of important applications. AQC remains, however, poorly understood theoretically and its mathematical underpinnings are yet to be satisfactorily identified. Through M-C theory, we bring a completely novel perspective that we hope opens the door for using such mathematics in the realm of quantum computations, providing a secure foundation for AQC. Specifically, we show that the singular homology of a certain cobordism, that we construct from the given Hamiltonian using the handlebody decomposition, defines the adiabatic evolution. Our result is based on E. Witten s famous construction for Morse homology that was derived in the very di erent context of supersymmetric quantum mechanics. We investigate how such topological invariants can be an obstruction to any computational speed-up in AQC (i.e., the scaling of the spectral gap). Particular attention is given to several examples exhibiting non-stoquastic behavior. A brief history of Morse theory The origins of Morse theory go back to the nineteen-thirties when M. Morse noted that important topological properties of smooth manifolds can be obtained from the critical points of smooth functions on them. To understand this, let M be a manifold with, possibly empty, boundaries M 0 and M 1 ;amorsetheoristwouldsaym is a cobordism from M 0 to M 1 and denote it by (M,@M), = M 0 [M 1. Figure 1 depicts two examples of cobordims with two di erent topologies. The one on the left (tea pot like) has an empty boundary (i.e., is a closed surface) and the one on the right (known as pair of pants) has a non empty boundary (M 0 is the lower circular boundary and M 1 is the disjoint union of the two upper circular boundaries). A point p in M is said to be acriticalpointofthereal-valuedfunctionf 2 C 1 (M) if the di erential map df (p) is identically zero. This, of course, amounts to equating the partial derivatives of f at p to zero, once a coordinates system is chosen. Geometrically, the tangent plane of M at 1

2 Figure 1: (Left) Deformed Sphere. The height function is Morse and has four critical points. (Right) The so-called pair of pants has a di erent topology (with Euler characteristic = 1, compared to 2 for the sphere). The height function is also Morse but has only one critical point. The pair of pants is the cobordism which we assign to Grover s search algorithm. the critical point p is horizontal. The critical point p is said to be non degenerate if the the determinant of the Hessian of f at p is not zero. If all the critical points of the function f are non degenerate then f is said to be a Morse function. In Figure 1 (left), the height function is Morse and has four non degenerate critical points: a minimum, a saddle point and two maxima. On the second surface, Figure 1 (right), the height function is also Morse, with only one critical point: a saddle point. M. Morse observation is that, with knowledge of critical points, it is possible to reverse engineer the original topology. The tool for that is the powerful handlebodies decomposition procedure which we use repeatedly in this work. Figure 2 provides the dictionary between the critical points and the handles that one can use to recover the cobordism on which the Morse function is defined. For instance, the surface on Figure 1(left)isrecoveredbyglueinga0-handle,1-handleandtwo2-handles,corresponding to the minimum, the saddle point and the two maxima respectively. The full power of Morse theory was unleashed by R.Thom and S. Smale (the latter with his work on the Poincaré s conjecture) in the sixties and subsequently, in the eighties, by E. Witten with his influential paper [17]. To the function f one assigns the (downward) gradient flow given by x 0 (t) = (rf) x(t) forsmoothcurvesx(t) 2 M. Trajectories (instantons) of a such gradient flow define a complex whose homology turns out to be isomorphic to the singular homology of the given manifold M. This is the apex result of Morse theory, known as Morse homology theorem, and plays a central role in this paper. (a) 0-handle (b) 1-handle (c) 2-handle Figure 2: Handles. 2

3 Main results The objective of the present work is to import the topological tools briefly introduced above to the context of adiabatic evolution: we would like to describe topologically, the adiabatic solutions of the Schrödinger equation (~ is set to 1): Hamiltonians of interest H(t) are of the form '(t)i = H(t) '(t)i. H(t) = initial (t)h initial + problem (t)h problem (0.2) with H initial and H problem non commuting Hamiltonians. The functions initial and problem are such that H(0) = H initial and H(T )=H problem for some finite time T [The usual smoothness assumptions on H(t) andh 0 (t) andtheirspectralprojectionsarealso assumed]. The first thing which needs to be done is to identify the relevant cobordism (M,@M) and the Morse function f 2 C 1 (M). It turns out that this is done first by finding the function f and then constructing (M,@M) asits domainofdefinition.achoicefor the function f is given by the real-valued (polynomial) function f(, t) =det (H(t) I). The energies of the Hamiltonian H(t) arenowlevelsets(contours)forthegradient flow defined by the negative of the gradient of f: 8 d >< ( ) f( ( ),s( )), d (0.3) >: d d s( ) sf( ( ),s( )). In fact, the level sets are uniquely defined as orthogonal curves to the flow lines. Assuming now that the critical points of the function f are non degenerate (in which case f is Morse) the cobordism (M,@M) is readily given by the handlebodies decomposition procedure from the set of critical points of f: Proposition. If f is Morse then the spectrum of H(t) corresponds to the level sets of the gradient flow given by f and defined on (M,@M). The cobordism (M,@M) is obtained from the critical points of f using the handlebodies decomposition procedure. Example (Grover s problem). The pair of pants in Figure 1 (Right) is the cobordism (M,@M) one gets for the search adiabatic algorithm [15] where H(t) =(1 t) I ˆ0ihˆ0 + t (I uihu ) with ui being the marked sought item in the large unsorted database { ii, i2 Z n }. The initial Hamiltonian I ˆ0ihˆ0 is written in the Hadamard basis while the problem Hamiltonian (I uihu ) is written the computational basis. Straightforward computations yield the characteristic polynomial of H(t) and the depicted pair of pants. 3

4 Once the cobordism This, noticeably, implies that the singular homology together with the set of critical points uniquely define the gradient flow and consequently, the adiabatic solutions of (0.1). Theorem. Let f(, t) =det(h(t) I) be Morse and let be the associated cobordism constructed in Proposition above. The adiabatic flow associated to H(t) is uniquely defined by the set critical(f) and the singular homology H (M,@M; K). Example (Grover s problem continued). Gauss-Bonnet theorem forces the Gaussian curvature K to distribute itself around the unique critical (beyond it, the curvature is zero). In fact, we have Z Z Kd Kd = 2 + O(x 2 ) (0.4) (M,@M) V (p) where x 2 =2 n and V (p) is an arbitrary small neighbourhood around the saddle point p. This shows that the curvature is very densely distributed around the saddle point. The exponential shrinking of the spectral gap for the search adiabatic algorithm is reminiscence of this fact. Let us finish this extended abstract by noting that the requirement that f is Morse can be relaxed. This is an upshot of Conley s generalization of Morse theory. Instantons connect now the isolated neighbourhoods of the critical points. Conley index at a critical point p generalizes Morse index and is given by the homotopy type of the isolated neighbourhood of p. ThisimpliesthatConleyindexremainsunchangedunder small perturbations around p. A key point is to choose perturbations that split the degenerate critical point p into a finite number of non degenerate points (confined inside the isolated neighbourhood of p). A typical simple example is the function f(, s) =s 3 3s 2 which is replaced with the Morse function f " (, s) =s 3 3s 2 + "s. Theorem 1 can be extended to the degenerate case following the same lines. References [1] D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd, and O. Regev. Adiabatic quantum computation is equivalent to standard quantum computation. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages42 51, Oct [2] J. E. Avron and A. Elgart. Adiabatic theorem without a gap condition. Communications in Mathematical Physics, 203(2): ,1999. [3] A. Banyaga and D. Hurtubise. Lectures on Morse homology, volume 29 of Kluwer Texts in the Mathematical Sciences. Kluwer Academic Publishers Group, Dordrecht, [4] M. Born and V. Fock. Beweis des adiabatensatzes. Zeitschrift für Physik, 51(3): ,

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