UCSD Spring School lecture notes: Continuous-time quantum computing

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1 UCSD Spring School lecture notes: Continuous-tie quantu coputing David Gosset 1 Efficient siulation of quantu dynaics Quantu echanics is described atheatically using linear algebra, so at soe level is very siple. On the other hand to siulate a quantu syste using a classical coputer sees to require storing and anipulating exponentially large vectors (in the size of the quantu syste). Motivated by this apparent difficulty, in 1982 Richard Feynan suggested that a quantu coputer could be used for siulating other quantu systes [Fey82]. Consider a quantu spin syste coposed of n qubits, which is described by a noralized vector ψ (C 2 ) n. The Schrodinger equation d ψ = ih ψ (1) dt describes the tie evolution of such a quantu syste with a Hailtonian H. The Hailtonian H ay itself depend on tie and later we will see an exaple of this. For now, let us specialize to the case in which H is tie-independent, and the solution to Eq. (1) is easily seen to be ψ(t) = e iht ψ(0). (2) Following Lloyd [Llo96], we now describe a siple ethod whereby a universal quantu coputer can efficiently siulate tie evolution Eq. (2) (below we ai for siplicity; later we suarize ore recent ethods which iprove the efficiency of the siulation). Quantu siulation reains one of the ost copelling applications in which quantu coputers achieve an exponential speedup over the best known classical algorith. Theore 1 ([Llo96]). There is an efficient classical algorith which takes as input a precision ɛ (0, 1), a tie t > 0, and a k-local Hailtonian H = h j. j=1 The algorith coputes a quantu circuit coposed of poly(, t, ɛ 1 ) gates such that the unitary U ipleented by the circuit satisfies U e iht ɛ. (3) 1

2 Exercise 2. Let A, B be n-qubit operators and δ (0, 1) such that A δ and B δ. Then e A e B e A+B O(δ 2 ). Exercise 3. Suppose V 1, V 2,..., V M and W 1, W 2,..., W M are unitary operators. Then V 1 V 2... V M W 1 W 2... W M M W j V j. j=1 Proof of Theore 1. Let R be a positive integer satisfying R t (recall is the nuber of local ters in H and t is the evolution tie). We start by dividing up the evolution tie into sall segents of length t/r: e iht = (e iht/r) R Now let us consider one segent. Below we will show Using this fact and Exercise 3 we get e iht/r e ih 1t/R e ih 2t/R... e ih t/r O( 3 t 2 R 2 ). (5) e iht ( e ih 1t/R e ih 2t/R... e ih t/r ) R O( 3 t 2 R 1 ). and so we can ake the RHS at ost ɛ by taking R = Θ( 3 t 2 ɛ 1 ). With this choice of R the quantu circuit U = (e ) ih1t/r e ih2t/r... e ih R t/r satisfies Eq. (3) and the above is a decoposition into R = poly(t,, ɛ 1 ) k-local gates. This establishes the theore since these k-local gates can be further decoposed into a universal set of 1- and 2-local gates while incurring only polylogarithic overhead in the size of the circuit. It reains to show Eq.(5). For each 0 L define Then U L = e i L j=1 h Lt/R e ih L+1 t/r e ih L+2t/R... e ih t/r. U L U L 1 = e i L j=1 hlt/r i L 1 e j=1 h jt/r e ih L t/r O(L 2 t 2 R 2 ). (6) where we used Exercise 2. Since U = e iht/r and U 0 = e ih 1t/R e ih 2t/R... e ih t/r we get e iht/r e ih 1t/R e ih 2t/R... e ih t/r = L=1 (U L U L 1 ) O( 3 t 2 R 2 ). (4) 2

3 While the above algorith establishes that efficient siulation of Hailtonian dynaics is possible, the circuit produced by the above algorith consists of O( 4 t 2 ɛ 1 ) gates which is far fro optial. In the roughly 20 years since Lloyd s paper there have been any new quantu algoriths for Hailtonian siulation which achieve better scaling in various paraeters. For exaple it is known that one can siultaneously achieve linear scaling in the evolution tie t (iproving the above by a quadratic factor) and a logarithic dependence on ɛ 1 (an exponential iproveent) [BCC + 17, BCK15, LC17]. 2 Adiabatic Quantu Coputation In this section we are going to iagine that we have a syste of qubits that evolves according to the Schrodinger equation with a local tie-dependent Hailtonian H which we get to choose. We will be interested in the case where the Hailtonian changes very slowly. Motivating exaple Adiabatic quantu coputing is a odel of coputation based on the adiabatic theore of quantu echanics. Roughly speaking, the adiabatic theore states that if a quantu syste is prepared in its ground state and then the Hailtonian is changed very slowly then the state of the syste will be well approxiated by the ground state of the instantaneous Hailtonian at all ties. To understand how adiabatic quantu coputation can be used to solve coputational probles, let s begin with a otivating exaple which was proposed in Ref. [FGGS00]. In that paper the authors introduced the adiabatic odel of quantu coputation and proposed the following quantu algorith for 3-satisfiability. Suppose we are given a 3-satisfiability instance on n bits with clauses. Define h(z) to be the function that coputes the nuber of clauses violated by a given bit string z {0, 1} n. We can write h(z) = h c (z) h c (z) = c=1 { 0 if z satisfies clause c 1 otherwise. Clearly we can solve 3-SAT if we can find the bit string z 0 such that h(z 0 ) is inial. Now define a Hailtonian H P which is diagonal in the coputational basis H P z = h(z) z z {0, 1} n. It is not hard to see that H P can be expressed as a 3-local Hailtonian: Exercise 4. Show that H P can be expressed as a 3-local Hailtonian. Its ground state is the coputational basis state z 0 and if we could prepare this state then we can solve the given 3-SAT instance. So our goal in the following will be to prepare the ground state of H P using a quantu algorith.. 3

4 Farhi et. al consider a one-paraeter faily of Hailtonians that interpolates between a beginning Hailtonian H B whose ground state is easy to prepare and the proble Hailtonian H P whose ground state provides the solution to the 3-SAT instance. H(s) = (1 s)h B + sh P 0 s 1 One possible choice for the beginning Hailtonian is H B = n j=1 (1 X j )/2 which describes a syste of non-interacting spins. The ground state of this Hailtonian is the product state + n where + = 2 1 ( ). An adiabatic quantu algorith for 3-satisfiability is as follows: 1. Prepare the ground state + n of H(0) = H B. 2. Slowly change the Hailtonian H(s) fro H(0) to H(1). In particular change the paraeter s(t) = t/t where t is tie and T is the total tie. 3. Measure all qubits in the coputational basis to obtain a bit string z {0, 1} n The adiabatic theore (desribed below) guarantees that if the total tie T is large enough then the final quantu state after step 2 will be well approxiated by the ground state z 0 of H(1) and the result of the final easureent will be z = z 0 which provides the solution to the given 3-SAT instance. The runtie of the above algorith using our adiabatic quantu coputer (which is capable of ipleenting Schrodinger tie evolution with a slowly varying Hailtonian) is equal to the total evolution tie T. To understand the efficiency of this algorith we need to understand how big T needs to be as a function of n in order to ensure that the state of the syste after step 2 is well approxiated by the ground state z 0 of H(1). A partial answer to this question is provided by the adiabatic theore which we describe in the next section. Before proceeding we ake two rearks. Firstly, tie evolution with a local Hailtonian can be siulated efficiently on a universal quantu coputer. In the previous section we describe a siple siulation algorith for the case of tie-independent Hailtonians but one can straightforwardly extend these ethods to the tiedependent case. This iplies that any algorith developed using adiabatic tie evolution can be ipleented with only polynoial overhead using a standard quantu coputer. Secondly, note that we don t expect quantu coputers to be able to solve NP-coplete probles such as 3-SAT in polynoial tie. We expect that the run tie T of the above algorith ust scale exponentially with n in the worst case. 4

5 The adiabatic theore and the runtie of adiabatic algoriths Let H(s) 0 s 1 be a sooth 1-paraeter faily of local Hailtonians. Suppose for siplicity that the ground state of H(s) is unique for all s [0, 1], and denote this ground state by φ(s). Iagine initializing the syste in the ground state φ(0) of H(0), and then evolving with a tie-varying Hailtonian H(s = t/t) where t is tie and T is the total evolution tie. The tie-evolved state ψ(t) is defined by d ψ = ih(t/t) ψ 0 t T. dt The adiabatic theore states that as T the tie-evolved state ψ(t = st) approxiates the ground state φ(s) of H(s). Below we quote a rigorous version of this stateent fro Ref. [JRS07] which provides an error bound for finite T. Write the eigenvalues of H(s) as E 0 (s) < E 1 (s) E 2 (s)... E 2 n(s) 0 s 1, where the leftost inequality is strict because we assue the ground state is unique. Define the iniu spectral gap g in = in 0 s 1 (E 1(s) E 0 (s)). Also define { C = ax 1, ax ds 0 s 1 dh } H, ax 0 s 1 d2 ds 2. (7) Theore 5 (Special case of Theore 3 fro Ref. [JRS07]). Suppose H(s) is sooth oneparaeter faily of local Hailtonians with a unique ground state for all 0 s 1 as described above. Then for all 0 s 1 we have ( ) ψ(t = st) φ(s) 2 1 O C 4 T 2 g 6 in where the big-o hides a universal constant which in particular does not depend on s. Fro the above we see that by choosing a total running tie T = Θ(C 2 g 3 in ) (8) we can ensure that at all interediate ties t there is a high overlap between the state of the syste and the ground state φ(t/t) of the instantaneous Hailtonian. Can we use Eq. (8) in practice to bound the runtie of quantu adiabatic algoriths (such as the one considered in the previous section)? Let us suppose we are interested only in whether a particular quantu adiabatic algorith is efficient, that is, we ai to understand whether or not Eq. (8) scales polynoially with n. In ost cases of interest including the exaple fro the 5

6 previous section one can easily upper bound C = O(poly(n)) and so the efficiency of the quantu adiabatic algorith is deterined by the iniu spectral gap g in. The algorith is efficient if g in is lower bounded by an inverse polynoial function of n and inefficient otherwise. Unfortunately, lower bounding the iniu spectral gap g in is a notoriously difficult proble and there are few exaples of nontrivial Hailtonians H(s) for which useful estiates have been obtained. Equivalence between adiabatic quantu coputation and the circuit odel We have discussed how, at least in principle, adiabatic evolution can be used to solve certain coputational probles. Moreover, such quantu coputations can be efficiently siulated by a universal quantu coputer. It was shown by Aharanov et al. [AVDK + 08] that any quantu circuit can also be efficiently siulated by adiabatic evolution. This establishes that the two odels are in fact equivalent. Theore 6 ([AVDK + 08]). A quantu circuit can be siulated efficiently by an adiabatic quantu coputation. Proof sketch. Suppose we are given a quantu circuit U = U()U( 1)... U(1) where each U(j) is a 1- or 2-qubit gate. The adiabatic quantu coputer will have two registers a data register with n qubits and a clock register with qubits. Define a 5-local Hailtonian H(s) = H input + I H clock + (1 s)(i ) + s H j j=1 where H input = H clock = n i=1 1 1 i (9) n i,i+1 (10) i=1 H j = 1 2( I j 1,j,j+1 + I j 1,j,j+1 U(j) j 1,j,j+1 U(j) j 1,j,j+1 ) (11) Eq. (11) should be odified when j = 1 or j = in the obvious way (by reoving operators which act on qubits j 1 and j + 1 respectively). Define states ĵ = (U(j)U(j 1)... U(1) 0 n ) 1 j 0 j j = 0, 1,...,. is When s = 0 the Hailtonian is diagonal in the coputational basis and its unique ground state 0 = 0 n data 0 clock. (12) 6

7 When s = 1 the Hailtonian is the Feynan-Kitaev construction encountered earlier and its unique ground state is ĵ. (13) j=0 To siulate the given quantu circuit by adiabatic evolution one prepares the initial product state 0, and then slowly changes the Hailtonian as a function of tie as H(t/T) for 0 t T. Here T is chosen large enough so that at the end of the evolution the state is well approxiated by the history state Eq. (13). A final easureent of all qubits of the clock register in the coputational basis gives outcoes 1 with probability 1/( + 1) in which case the state of the data register contains the output of the quantu coputation U()U( 1)... U(1) 0 n. It reains to show that T can be chosen to scale polynoially with n and. It is convenient to use the fact that for all 0 s 1 the Hailtonian H(s) preserves the subspace A = span{ 0,..., }. (14) Since the state of our adiabatic quantu coputer is initialized in 0 A, and since H(s) preserves A, the entire coputation proceeds within this subspace. We can therefore consider A to be the whole Hilbert space and apply the adiabatic theore to the restricted Hailtonian H(s) A. This restricted Hailtonian is an atrix with a siple one diensional structure: H(s) A = (1 s)diag(0, 1, 1,... 1) + s (15) Note that the above atrix does not depend on the quantu circuit at all. Its iniu spectral gap on the interval s [0, 1] can be bounded as g in Ω( 2 ) (see Ref. [AVDK + 08] for details). We ay also straightforwardly upper bound the paraeter C since and d 2 ds 2 (H(s) A) = 0 d ds (H(s) A) 1 + ( + 1) where we used the triangle inequality and Eq. (15). Plugging C = O() and g in Ω( 2 ) into Eq. (8) we see that a total evolution tie T = O(poly()) suffices to siulate the quantu circuit using an adiabatic quantu coputer. 7

8 References [AVDK + 08] Dorit Aharonov, Wi Van Da, Julia Kepe, Zeph Landau, Seth Lloyd, and Oded Regev. Adiabatic quantu coputation is equivalent to standard quantu coputation. SIAM review, 50(4): , [BCC + 17] [BCK15] [Fey82] Doinic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D Soa. Exponential iproveent in precision for siulating sparse hailtonians. In Foru of Matheatics, Siga, volue 5. Cabridge University Press, Doinic W Berry, Andrew M Childs, and Robin Kothari. Hailtonian siulation with nearly optial dependence on all paraeters. In Foundations of Coputer Science (FOCS), 2015 IEEE 56th Annual Syposiu on, pages IEEE, Richard P Feynan. Siulating physics with coputers. International journal of theoretical physics, 21(6-7): , [FGGS00] Edward Farhi, Jeffrey Goldstone, Sa Gutann, and Michael Sipser. Quantu coputation by adiabatic evolution. arxiv preprint quant-ph/ , [JRS07] [LC17] Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler. Bounds for the adiabatic approxiation with applications to quantu coputation. Journal of Matheatical Physics, 48(10):102111, Guang Hao Low and Isaac L Chuang. Optial hailtonian siulation by quantu signal processing. Physical review letters, 118(1):010501, [Llo96] Seth Lloyd. Universal quantu siulators. Science, pages ,