On the gap of Hamiltonians for the adiabatic simulation of quantum circuits
aa r X i v : . [ qu a n t - ph ] J u l LA-UR-13-23134 SAND 2013-5836J
On the gap of Hamiltonians for the adiabatic simulation of quantum circuits
Anand Ganti ∗ Sandia National LaboratoriesAlbuquerque, New Mexico 87185, USA
Rolando D. Somma † Los Alamos National LaboratoryLos Alamos, New Mexico 87545, USA (Dated: June 26, 2018)The time or cost of simulating a quantum circuit by adiabatic evolution is determined by thespectral gap of the Hamiltonians involved in the simulation. In “standard” constructions based onFeynman’s Hamiltonian, such a gap decreases polynomially with the number of gates in the circuit, L . Because a larger gap implies a smaller cost, we study the limits of spectral gap amplification inthis context. We show that, under some assumptions on the ground states and the cost of evolvingwith the Hamiltonians (which apply to the standard constructions), an upper bound on the gapof order /L follows. In addition, if the Hamiltonians satisfy a frustration-free property, the upperbound is of order /L . Our proofs use recent results on adiabatic state transformations, spectralgap amplification, and the simulation of continuous-time quantum query algorithms. They alsoconsider a reduction from the unstructured search problem, whose lower bound in the oracle costtranslates into the upper bounds in the gaps. The impact of our results is that improving the gapbeyond that of standard constructions (i.e., /L ), if possible, is challenging. I. INTRODUCTION
Adiabatic quantum computing (AQC) is an alternativeto the standard circuit model of quantum computation.In AQC, the input is a (qubit) Hamiltonian H (1) andthe goal is to prepare the ground state of H (1) by meansof slow or adiabatic evolutions. One then sets an ini-tial Hamiltonian H (0) and builds a Hamiltonian path H ( g ) , ≤ g ≤ , that interpolates between H (0) and H (1) . If the ground states of H ( g ) are continuously re-lated and remain at a spectral gap of order ∆ with anyother eigenstate during the evolution, the quantum adi-abatic approximation implies that, for ˙ g ( t ) ≤ ǫ ∆ q h , (1)the ground state of H (1) can be adiabatically preparedwith fidelity − ǫ . < q ≤ and h depends on k ∂ n H ( g ) /∂g n k q +1 , n = 1 , , for differentiable paths [1–5].A key feature of AQC is that it constitutes a “natural”model for problems that efficiently reduce to the compu-tation of ground-state properties. Some of these are prob-lems in combinatorial optimization [6–12] and problemsin many-body physics, e.g. the computation of a quan-tum phase diagram [13]. Whether AQC is robust to de-coherence or not is unclear and a complete fault-tolerant ∗ Electronic address: [email protected] † Electronic address: [email protected] implementation of AQC remains unknown [14, 15]. Nev-ertheless, the role of the spectral gap is imperative in anoisy implementation of AQC: a bigger ∆ could implya smaller running time [Eq. (1)] and a reduction of the(unwanted) population of excited states due to thermaleffects. Our goal is then to study the limits and possi-bilities of amplifying the gap in AQC. Roughly stated,we are addressing the following question: Given H ( g ) with gap ∆( g ) and ground state | ψ ( g ) i , can we find ˜ H ( g ) with gap ˜∆( g ) ≫ ∆( g ) and ground state satisfying | ˜ ψ ( g ) i ≈ | ψ ( g ) i ? Our motivation is the same as that ofRef. [16]. We are particularly interested in amplifyingthe gap of those Hamiltonians that arise in the adiabaticsimulation of general quantum circuits– see below.The power of AQC and the standard quantum cir-cuit model are equivalent. That is, any algorithm inthe AQC model with a running time T , that preparesa quantum state | ψ (1) i , can be simulated with a quan-tum circuit of L ∈ poly( T ) unitary gates that preparesa sufficiently close state to | ψ (1) i when acting on sometrivial initial state [17–20]. The converse also holds: Anyquantum circuit of L unitary gates that prepares a quan-tum state (cid:12)(cid:12) φ L (cid:11) , when acting on some trivial initial state,can be simulated within the AQC model by evolvingadiabatically with suitable Hamiltonians H ( g ) for time T ∈ poly( L ) . The ground state of the final Hamiltonian, | ψ (1) i , has a large probability of being in (cid:12)(cid:12) φ L (cid:11) after asimple measurement [21–23]. H ( g ) depends on the uni-taries that specify the quantum circuit.To describe our results in detail, we review the first“standard” construction in Ref. [21], which is based onFeynman’s Hamiltonian [24]. U is a quantum circuitacting on n qubits that prepares the “system” state (cid:12)(cid:12) φ L (cid:11) = U L . . . U (cid:12)(cid:12) φ (cid:11) , after the action of L unitary gates U , . . . , U L . There is also an ancillary system, denoted by“clock”, whose basis states are {| i c , | i c , . . . , | L i c } . The(final) Hamiltonian H U is mainly a sum of two terms.The first term is the so-called Feynman Hamiltonian: H U Feynman = L X l =1 h U ,l ,h U ,l = 12 (1l ⊗ | l ih l | c + 1l ⊗ | l − ih l − | c − (2) − U l ⊗ | l ih l − | c − ( U l ) † ⊗ | l − ih l | c ) . is the trivial operation on the system’s state. H U Feynman can be easily diagonalized by using the Fourier transform.The eigenvalues are − cos k , with k = 2 πm/ ( L + 1) and m ∈ Z . Then, the lowest eigenvalue is zero ( k = 0 )and the gap is of order /L [ k = 2 π/ ( L + 1) for thesmallest nonzero eigenvalue]. Each eigenvalue appearswith multiplicity n , corresponding to each state | σ i ofthe system. The eigenstates of H U Feynman are √ L + 1 L X l =0 e ikl U l . . . U | σ i ⊗ | l i c , (3)where U = 1l . If | σ i = (cid:12)(cid:12) φ (cid:11) , the eigenstate in Eq. (3) hasprobability / ( L + 1) of being in the state output by thecircuit. That is, we can prepare (cid:12)(cid:12) φ L (cid:11) with such a proba-bility by a projective measurement of the clock register onthe state of Eq. (3). We can remove the multiplicity of thelowest eigenvalue if we add a second term, H input , whoseexpected value vanishes when | σ i = (cid:12)(cid:12) φ (cid:11) and is strictlypositive otherwise. For example, if (cid:12)(cid:12) φ (cid:11) = | + i ⊗ n , where | + i = ( | i + | i ) / √ , H input in Ref. [21] corresponds to H input = n X j =1 |−ih−| j ⊗ | ih | c , with |−i = ( | i − | i ) / √ . In this case, H input sets a“penalty” if the system-clock initial state is different from | + i ⊗ n ⊗ | i c . The lowest eigenvalue of H input is zero andthe gap is a constant independent of L (i.e., ∆ input = 1 ).Then, the Hamiltonian H U = H U Feynman + H input (4)has (cid:12)(cid:12) ψ U (cid:11) = 1 √ L + 1 L X l =0 (cid:12)(cid:12) φ l (cid:11) ⊗ | l i c (5)as unique ground state [ k = 0 in Eq. (3)], where (cid:12)(cid:12) φ l (cid:11) = U l . . . U (cid:12)(cid:12) φ (cid:11) . We will refer to (cid:12)(cid:12) ψ U (cid:11) as the “historystate”. The lowest eigenvalue of H U is also zero andthe spectral gap satisfies ∆ U ∈ Θ(1 /L ) [25]. It is sim-ple to construct an interpolating path H U ( g ) that has a spectral gap ∆ U ( g ) = ∆ U ∈ Θ(1 / poly L ) for all g and H U (1) = H U . This is done by, for example, parametriz-ing the unitaries in the circuit so that U l → U l ( g ) in Eq. (4), and U l (0) = 1l , U l (1) = U l . Then, theground state (cid:12)(cid:12) ψ U (1) (cid:11) = (cid:12)(cid:12) ψ U (cid:11) can be prepared from (cid:12)(cid:12) ψ U (0) (cid:11) = (cid:12)(cid:12) φ (cid:11) ⊗ P l | l i c / √ L + 1 by evolving adiabat-ically with H U ( g ) for time T ∈ O [poly( L )] [see Eq. (1)]. H U is often regarded as “unphysical” as the system-clock interactions may represent non-local interactionsof actual quantum subsystems (qubits). Then, a num-ber of steps that include modifications of the gates inthe circuit and techniques from perturbation theory (e.g.,perturbation gadgets) [26], allow us to reduce H U to aphysical, local Hamiltonian H U local . Such steps preservethe two main ingredients for showing the equivalence be-tween AQC and the circuit model: i- that the spectralgap of the local Hamiltonian, ∆ U local , is bounded from be-low by / poly( L ) and ii- that the ground state has suf-ficiently large probability of being in (cid:12)(cid:12) φ L (cid:11) after a simplequantum operation (e.g., a simple projective measure-ment). It is important to remark that ∆ U local is smallerthan ∆ U in standard constructions [21, 23]. For this rea-son, some attempts to improve the running time of theadiabatic simulation of a quantum circuit consider firstthe amplification of ∆ U by making simple modificationsto H U (see Ref. [27] for an example); Our results concernthe amplification of ∆ U .In this report we show that, under some assumptionson the ground states and the time or cost of evolving withthe Hamiltonians, an upper bound on the gap of order /L follows. Furthermore, if the Hamiltonians addition-ally satisfy a so-called frustration-free property, then theupper bound is /L . An implication of our results is thatsimple modifications to H U in Eq. (4) are not sufficient toamplify its gap. While such modifications could be usefulto prepare the desired state via a constant-Hamiltonianevolution [28], they may not be useful to prepare the stateadiabatically. Our proofs are constructive, i.e., we finda reduction from the unstructured search problem [29](Sec. II), whose lower bound on the oracle cost [30] (i.e.,the number of queries to the oracle needed) can be trans-formed into the upper bounds on the gaps (Sec. III).Clearly, the only way to obtain a bigger gap, if gap am-plification is indeed possible, is by avoiding one or moreassumptions needed for our proofs. This suggests a mi-gration from those constructions that are based on Feyn-man’s Hamiltonian. II. SEARCH BY A GENERALIZEDMEASUREMENT-BASED METHOD
The proof of an upper bound on ∆ U uses a reduc-tion from the unstructured search problem or SEARCH.In this section, we show a quantum method that solvesSEARCH using measurements.For a system of n qubits, we let N = 2 n be the dimen-sion of the associated state (Hilbert) space H . Given anoracle O X , where the input X is a n -bit string, the goalof SEARCH is to output X . In quantum computing, O X implements the following unitary operation: O X | Y i = (cid:26) | Y i if X = Y , − | X i if X = Y .O X acts on H . A quantum algorithm for SEARCH uses O X and other X -independent operations to prepare astate sufficiently close to | X i . Thus, a projective mea-surement on this state outputs X with large probability.The (oracle) cost of the algorithm is given by the num-ber of times that O X is used. A lower bound Ω( √ N ) for the cost of SEARCH is known [30] and the famousGrover’s algorithm solves SEARCH with L/ ∈ Θ( √ N ) oracle uses [29]. Grover’s algorithm, denoted by U X ,is a sequence of two unitary operations, O X and R ,where R is a reflection over the equal superposition state (cid:12)(cid:12) φ (cid:11) = √ N P Y | Y i = | + i ⊗ n . The initial state is also (cid:12)(cid:12) φ (cid:11) . The state output by U X is (cid:12)(cid:12) φ LX (cid:11) and satisfies, inthe large N limit, (cid:13)(cid:13)(cid:12)(cid:12) φ LX (cid:11) − | X i ] (cid:13)(cid:13) ≪ . (6)There are other quantum methods that solveSEARCH, with optimal cost, using measurements. Onesuch method, first introduced in Ref. [31], involves twoprojective measurements: After preparing (cid:12)(cid:12) φ (cid:11) , a mea-surement of | ψ X i ≈ [ | X i + (cid:12)(cid:12) φ (cid:11) ] / √ , followed by a mea-surement of | X i , outputs X with probability close to / . The cost of this measurement-based method is dom-inated by the simulation of the first measurement. Sucha simulation can be done using the phase estimation al-gorithm [32] or by phase randomization [33]. Both meth-ods require evolving with a Hamiltonian that has | ψ X i aseigenstate. The evolution time is proportional to the in-verse gap of the Hamiltonian, which is needed to resolvethe desired state from any other eigenstate.Generalizations of the above measurement-basedmethod, that consider simulating measurements in otherstates, also solve SEARCH. To see this, we let | ζ X i ∈ H ′ and | ν i ∈ H ′′ be two pure quantum states that do anddo not depend on X , respectively. The correspondingHilbert spaces satisfy H ′′ ⊆ H ′ ⊆ H . We also define p ν,ζ X = tr[ h ζ X | ν ih ν | ζ X i ] ,p X,ζ X = tr[ h φ LX | ζ X ih ζ X | φ LX i ] , which are the probabilities of projecting | ν i into | ζ X i ,and | ζ X i into | X i , respectively, after a measurement (onthe corresponding Hilbert spaces). A generalization ofthe measurement-based method is described in Table I.The probability of success is p s ≥ p ν,ζ X · p X,ζ X . Table I. Generalized measurement-based method i- Prepare | ν i ii- Measure | ζ X i iii- Measure | X i We now obtain the time T of solving SEARCH (withprobability p s ) with the generalized measurement-basedmethod. We let G X be the Hamiltonian that has | ζ X i as unique ground state and the corresponding spectralgap of G X is ∆ U X . G X acts on the Hilbert space H ′ anddepends on O X . T is determined by the total time of evo-lution with G X needed to simulate the measurement of | ζ X i in step ii. Using the phase estimation algorithm [32]or evolution randomization [33], this time is T = c/ ∆ U X , (7)for some constant c ≥ π . The lower bound in the oraclecost of SEARCH can then be used to set a lower bound on T or, equivalently, an upper bound in ∆ U X . This resultsfrom noting that the evolution under G X can be wellapproximated with a discrete sequence of unitaries thatcontains O X . Nevertheless, to make a rigorous statementon ∆ U X , some assumptions on G X and the ground stateare needed. We provide such assumptions and our mainresults in the next section. III. GAP BOUNDS
We list three assumptions on G X and its ground state, | ζ X i . Assumption 1 : p ν,ζ X ∈ Θ(1) ∀ X .
That is, there exists an X -independent state | ν i thatcan be projected into | ζ X i , with high probability, after ameasurement. Assumption 2 : p X,ζ X ∈ Θ(1) ∀ X .
That is, | ζ X i can be projected into | X i , with high prob-ability, after a measurement. Assumptions 1 and 2 re-sult in a probability of success p s ∈ Θ(1) when solv-ing SEARCH with the generalized measurement-basedmethod of Table I.Assumptions 1 and 2 may be combined into one asdescribed in Appendix A. Also, a generalization of As-sumption 2 to any circuit U is a requirement of having aground state with large probability of being in the stateoutput by the circuit after measurement. This propertyis desired for Hamiltonians involved in the adiabatic sim-ulation of quantum circuits. Assumption 3 : For all t ∈ R and fixed ǫ , ≤ ǫ < ,there exists a unitary operation W X = ( S. ˜ O X ) r , where S is also a unitary operation that does not depend on X , ˜ O X = O X ⊗ is the oracle for SEARCH acting on thelarger Hilbert space H ′ , r ≤ | c ′ t | γ , and k e iG X t − W X k ≤ ǫ .c ′ > and γ ≥ are constants.Assumption 3 implies that the evolution operator de-termined by G X can be approximated, at precision ǫ ,by a sequence of unitary operations that uses the oracleorder | c ′ t | γ times. For some specific G X , such an approx-imation may follow from the results in Refs. [17–19] onHamiltonian simulation (see Sec. IV). Theorem. If G X and | ζ X i satisfy Assumptions 1, 2,and 3, ∆ U X ∈ O (1 /L /γ ) . In addition, if G X satisfies a frustration-free property [34,35], ∆ U X ∈ O (1 /L /γ ) . The definition of a frustration-free Hamiltonian is in-cluded in the proof. The second bound applies underan additional requirement on G X . This requirement to-gether with the constants for the upper bounds are alsodiscussed in the proof.We note that the gap in the second upper bound maynot be the “relevant” gap for the adiabatic simulation.In certain cases, for example, the adiabatic simulationmay not allow for transitions from the ground state tothe first-excited state due to symmetry reasons. Never-theless, the first bound still holds for the relevant gap inthese cases. Proof.
Simulating the measurement in step ii of thegeneralized measurement-based method requires an evo-lution time T = c/ ∆ U X [Eq. (7)]. From Assumption 3,the evolution can be approximated by a quantum circuitthat uses the oracle r times, with r ≤ ( c ′ T ) γ . The lowerbound on the cost of solving SEARCH [30] implies (cid:18) c ′ c ∆ U X (cid:19) γ ≥ r ≥ α √ N ≥ α L , where α > is a constant because p s ∈ O (1) . Then, ∆ U X ≤ c ′ c/ (2 αL ) /γ . G X is a frustration-free Hamiltonian if it is a sum ofpositive semidefinite terms and the ground state | ζ X i isa ground state of every term [34–36]. In this case, it ispossible to preprocess G X and build a Hamiltonian ˜ G X that has | ˜ ζ X i = | ζ X i ⊗ | i a (8)as (unique) eigenstate of eigenvalue zero, where | i a de-notes some simple, X -independent state of an ancillarysystem a. The corresponding spectral gap of ˜ G X for this state is ˜∆ U X ≥ √ ∆ U X – see Ref. [35] for detailson spectral gap amplification. Then, SEARCH can besolved with probability p s ∈ O (1) , using the general-ized measurement-based method, by evolving with ˜ G X for time T = c/ √ ∆ U X [37]. If Assumption 3 also appliesfor approximating the evolution operator e − i ˜ G X t , then √ ∆ U X ≤ c ′ c/ (2 αL ) /γ . This completes the proof. Corollary. If γ = 1 , then ∆ U X ∈ O (1 /L ) . Inaddition, if G X is frustration free as explained above, ∆ U X ∈ O (1 /L ) .It is possible to achieve γ → for some G X (seeSec. IV). Corollary.
If the eigenvalues of H U do not depend on U , the upper bounds on ∆ U X are upper bounds on ∆ U . IV. DISCUSSION: VALIDITY OF THEASSUMPTIONS AND IMPLICATIONS
We review the validity of the assumptions and implica-tions for some constructions found in the literature. Thefirst is the standard construction in Ref. [21], also dis-cussed in Sec. I. In this case, we consider a modificationof Grover’s algorithm so that U X = 1l L/ ( RO X ) L/ L/ ,with L ∈ Θ( √ N ) and the trivial (identity) operation.Such a modification is unnecessary but it simplifies theanalysis below. The state output by the modified circuitis unchanged; the only change is in the Hamiltonians. Asbefore, we let G X = H U X be the Hamiltonian associatedwith U X and | ζ X i = (cid:12)(cid:12) ψ U X (cid:11) be its ground state [i.e., thehistory state of Eq. (5) with U = U X ]. For the modi-fied circuit, the ground state has large overlap with the X -independent state | ν i = (cid:12)(cid:12) φ (cid:11) ⊗ p L/ L/ X l =0 | l i c . Similarly, | ζ X i has large overlap with the state | X i ⊗ p L/ L X l =3 L/ | l i c , because | X i ≈ (cid:12)(cid:12) φ LX (cid:11) [see Eq. (6)]. These Eqs. imply p ν,ζ X ≈ / and p X,ζ X ≈ / , so that Assumptions 1and 2 are readily satisfied. To study Assumption 3, wewrite G X = − O X ⊗ X l : U l = O X [ | l ih l − | c + | l − ih l | c ] + . . . = O X ⊗ P c + H s − c . (9) P c is a Hamiltonian acting on the clock register that isa sum of commuting terms like | l ih l − | c + | l − ih l | c :the oracles O X are interleaved with the operations R inGrover’s algorithm. Then, the eigenvalues of P c are ± and k P c k ≤ , where k . k is the operator norm. H s − c is a system-clock Hamiltonian that does not depend on X : H s − c is a sum of H input and those terms in H U X Feynman that do not depend on O X . Using the results in Ref. [19],the operator exp { iG X t } can be well approximated us-ing O ( | t | log | t | ) oracles O X (see Appendix B). Thus, As-sumption 3 is satisfied for the construction of Ref. [21]and γ → assymptotically.To prove that G X is frustration free, we note that G X = W ( U X ) H W ( U X ) † , (10)where H is the Hamiltonian of Eq. (4) for the trivialcircuit and W ( U X ) = L X l =0 U l ⊗ | l ih l | c (11)is a unitary operation. For the modified Grover’s algo-rithm, U l ∈ { , R, O X } . It is simple to verify that (cid:12)(cid:12) ψ (cid:11) ∝ (cid:12)(cid:12) φ (cid:11) P l | l i c , h ,l (cid:12)(cid:12) ψ (cid:11) = 0 , h ,l ≥ , H input (cid:12)(cid:12) ψ (cid:11) = 0 , H input ≥ . This implies that H is frustration free andso are G X and H U for any U . Then, there exists ˜ G X = W ( U X ) ˜ H W ( U X ) † whose ground state is | ˜ ζ X i = | ζ X i⊗ | i a and whose gap is √ ∆ [38]. a is an ancilliary system of dimension L + n . Theoperators h U ,l have eigenvalues , and √ h U ,l = h U ,l .Then, from the results in Ref. [35], Sec. IV, we obtain ˜ G X = ˜ H U X Feynman + ˜ H input , (12)with ˜ H U X Feynman = L X l =1 h U X ,l ⊗ [ | l ih | a + | ih l | a ] , ˜ H input = n X j =1 |−ih−| j ⊗ | ih | c ⊗⊗ [ | L + j ih | a + | ih L + j | a ] . When U l = O X in the modified Grover’s algorithm, h U X ,l = 12 [1l( ⊗| l ih l | c + | l − ih l − | c )++ O X ⊗ | l i h l − | c + | l − i h l | c ] . Thus, another representation for ˜ G X is ˜ G X = O X ⊗ ˜ P c − a + ˜ H s − c − a , (13)with k ˜ P c − a k ≤ because k ˜ H U Feynman k ≤ (see Ap-pendix C). The system-clock-ancilla Hamiltonian ˜ H s − c − a is independent of X . Then, the evolution operator e i ˜ G X t can be approximated from the results in Ref. [19] us-ing the oracle O ( | t | log | t | ) times and the gadget in Ap-pendix B. It follows that γ → asymptotically for this case as well, and the gap satisfies ∆ U X ∈ ˜ O (1 /L ) . (The ˜ O notation accounts for the additional logarithmic fac-tor.) This upper bound is also valid for any ∆ U , becausethe eigenvalues of H U do not depend on U [Eq. (10)]. Ourresult is compatible with the lower bound on ∆ U obtainedin Ref. [21] (see Sec. I). It proves that our technique toestablish limits in the gap is effective. Nevertheless, aswe show below, our technique is powerful when analyzingthe gaps of Hamiltonians that are simple modificationsto the G X above, where obtaining the spectrum directlycan be challenging. We note again that, since the localHamiltonian constructed in Ref. [21] has a smaller gapthan that of H U or G X , the bound on the gap of G X translates into a bound on the gap of the local Hamilto-nian.We use the previous analysis to show a more gen-eral result. Consider a general Hamiltonian H U = W ( U ) H W ( U ) † for the adiabatic simulation of a quan-tum circuit, which uses a clock register, and whoseground state is of the form (cid:12)(cid:12) ψ U (cid:11) = W ( U ) (cid:12)(cid:12) ψ (cid:11) = L X l =0 α l (cid:12)(cid:12) φ l (cid:11) ⊗ | l i c , (14)and (cid:12)(cid:12) ψ (cid:11) = (cid:12)(cid:12) φ (cid:11) ⊗ P l | l i c . With no loss of generality, wecan assume that there exists l such that L X l = l | α l | ∈ Θ(1) . (15)If this condition is not satisfied, we can always applyan operation that permutes the clock states or we canadd trivial operations to the circuit so that Eq. (15) issatisfied (the spectrum of H U is unchanged). We let l be the largest l to satisfy Eq. (15). Then, we consider amodification of Grover’s algorithm so that U X = | ih | b ⊗
1l + | ih | b ⊗ (cid:16) l . ( RO X ) L − l (cid:17) , where b is an ancillary qubit (see Appendix A). L ∈ Θ( √ N ) . Basically, the modified Grover’s algorithm actstrivially, if the state of an ancillary qubit is | i b , or imple-ments the original Grover’s algorithm, if the state of theancilla is | i b . The initial state is | + i b ⊗ (cid:12)(cid:12) φ (cid:11) , and (cid:12)(cid:12) φ (cid:11) isthe equal superposition state as required in Grover’s al-gorithm. Assumptions 1 and 2 then follow from Eq. (15),for those ground states that can be described by Eq. (14).Additionally, if the Hamiltonian associated with U X canbe represented as in Eq. (9), with k P c k ≤ , the evolu-tion under G X can be well approximated using O ( t log t ) oracles and the upper bound on ∆ U X is of ˜ O (1 /L ) . SuchHamiltonians include those H ′U arising from modifiedFeynman Hamiltonians, where H ′U Feynman = P l β l h U ,l , | β l | ≤ , and those Hamiltonians that have an additionalterm H pointer = X l E l . ⊗ | l ih l | c , that acts solely in the clock space.For those H ′U , the spectrum is independent of U (i.e., ∆ U X = ∆ U ) and, in particular, G X = W ( U X ) H ′ W ( U X ) † [see Eqs. (10) and (11)]]. The unitaries U l involved in thedefinition of W ( U X ) are U l ∈ { , | ih | b ⊗
1l + | ih | b ⊗ O X , | ih | b ⊗ | ih | b ⊗ R } , for the current U X . H ′ actstrivially in the system and has tridiagonal form in thebasis {| i c , . . . , | L i c } . Then, with no loss of generality,we can assume that H is frustration free [39]. It followsthat G X is also frustration free and we can build ˜ G X = W ( U X ) ˜ H W † ( U X ) , by using the results of Ref. [35]. Because ˜ H is alsotridiagonal in the basis {| i c , . . . , | L i c } , ˜ G X admits arepresentation of the form of Eq. (13) in this case, with k ˜ P c − a k ≤ . Then, the oracle cost of simulating ˜ G X fortime t is also O ( t log t ) . This implies that, for modifiedFeynman Hamiltonians, the second bound on the gap ap-plies with γ → , and ∆ U ∈ ˜ O (1 /L ) .A few remarks are in order. First, we note that theabove result contradicts a statement in Ref. [27] claimingthat the gap can be amplified to order /L by including aterm of the form H pointer . Second, that an upper boundon ∆ U of order /L /γ is obtained when the Hamiltoniansatisfies the frustration free property, it does not contra-dict that, for some Hamiltonians, the “relevant” gap incertain subspace (e.g., the translationally invariant sub-space) may be larger. Nevertheless, such a relevant gapshould be limited by the bound on ∆ U obtained withoutassuming the property in the frustration (i.e., /L /γ inthis case). A third remark concerns the applicability ofour results to those constructions in which the Hamilto-nians are associated with one-dimensional quantum sys-tems, such as the one in Ref. [23]. These constructionswould require “breaking” the oracle O X into local, two-qubit pieces. While Assumptions 1 and 2 are easy toverify, a new version of Assumption 3 is required for thiscase. Such a version may be possible even if the oracle isnow a composition of two-qubit local operations, becausethe evolution operator with the one-dimensional Hamil-tonian may “reconstruct” a full oracle after certain unitof evolution time. However, we do not have any rigorousresult for this case and finding other suitable versions ofAssumption 3 is work in progress. Finally, Assumptions1 and 2 do not apply to the construction in A. Mizel,e-print: arXiv:1002.0846 (2010). V. ACKNOWLEDGEMENTS
We thank S. Boixo, R. Blume-Kohout, D. Gossett, A.Landahl, and D. Nagaj for insightful discussions. Weacknowledge support from the Laboratory Directed Re-search and Development Program at Sandia National Laboratories. Sandia National Laboratories is a multi-program laboratory managed and operated by SandiaCorporation, a wholly owned subsidiary of LockheedMartin Corporation, for the U.S. Department of Energy’sNational Nuclear Security Administration under contractDE-AC04-94AL85000.
Appendix A: More on Assumptions 1 and 2
In general, Assumption 2 is mostly an statement aboutthe ground state of the Hamiltonian H U that simulatesa quantum circuit, (cid:12)(cid:12) ψ U (cid:11) . Ideally, such state has largeprobability of being in the state output by the circuit, (cid:12)(cid:12) φ L (cid:11) ; that is, Pr( φ L | ψ U ) = Tr[ (cid:10) φ L (cid:12)(cid:12) ψ U ih ψ U (cid:12)(cid:12) φ L (cid:11) ] ∈ Θ(1) . We can then consider a modified quantum circuit thatuses an additional ancilla b prepared in | + i b so that itapplies the unitary U (original circuit) controlled on thestate | i of the ancilla, or does nothing otherwise. If wedenote the modified circuit by ¯ U , the output state is (cid:12)(cid:12) ¯ φ L (cid:11) = ¯ U (cid:0) | + i ⊗ (cid:12)(cid:12) φ (cid:11)(cid:1) = 1 √ | i b ⊗ (cid:12)(cid:12) φ (cid:11) + | i b ⊗ (cid:12)(cid:12) φ L (cid:11) ] . In this way, if the ground state of H U is a superpositionof system-clock states of the form (cid:12)(cid:12) φ l (cid:11) ⊗ | l i c , the groundstate of H ¯ U will be a superposition of states of the form (cid:12)(cid:12) ¯ φ l (cid:11) ⊗ | l i c , with (cid:12)(cid:12) ¯ φ l (cid:11) = ¯ U l · · · ¯ U ( | + i b ⊗ (cid:12)(cid:12) φ (cid:11) ) . When U = U X corresponds to Grover’s algorithm, if | ψ ¯ U i haslarge probability of being in | ¯ φ L i after measurement, thenit has large probability of being in both, (cid:12)(cid:12) φ (cid:11) and (cid:12)(cid:12) φ L (cid:11) ,after respective measurements. Since (cid:12)(cid:12) φ (cid:11) is independentof X , | ¯ φ L i satisfies Assumption 1 and 2 simultaneously.Thus, in Grover’s algorithm, Assumptions 1 and 2 canbe combined into a single one for Hamiltonians whoseground states are superpositions of (cid:12)(cid:12) φ l (cid:11) ⊗ | l i c . The gapbounds will apply to H ¯ U X in this case. Appendix B: Oracle simulation of the FeynmanHamiltonian associated with Grover’s algorithm
Following Ref. [19], the first step is to use the Trotter-Suzuki approximation that, in the case of the evolutionunder G X = O X ⊗ P c + H s − c , it yields terms of the form e − isO X ⊗ P c (B1)for some small s ∈ R . The goal in this section is topresent gadget that implements Eq. (B1) (i.e., a frac-tional oracle) using O X . Then, the problem is reducedto the one analyzed in Ref. [19], for which the oracle costis known.First, we note that there exists a unitary operation V c such that V c e − isO X ⊗ P c V † c = e − isO X ⊗ D c , (B2)where D c is a diagonal operator acting on the clock reg-ister, i.e., D c = X k λ k | k ih k | c , and | λ k | ≤ because k P c k ≤ . V c commutes with O X and it does not depend on X . The “gadget” of Fig. 1 usesthis observation to implement the operation of the rhs ofEq. (B2). Then, the desired operator of Eq. (B1) can beimplemented by conjugating the circuit of Fig. 1 with V c .This has to be compared with Fig. 3 of Ref. [19]. | k i c • • | k i c | i b R • R FE ✌✌ O X FIG. 1: Simulation of exp {− isO X ⊗ D c } [Eq. (B2)]. b isan ancilla qubit. The controlled operations are: R | i b ∝ p cos( sλ k / | i b − i p sin( sλ k / | i b and R | i b ∝ p cos( sλ k / | i b − (+) p sin( sλ k / | i b (see Fig. 3 inRef. [19]). The ancilla is measured at the end and the simula-tion of e − isλ k O X ⊗| k ih k | c succeeds if the outcome is | i b . Theoracle is controlled in the state | i b . If we use the simulation of Fig. 1 in the scheme shownin Fig.4 of Ref. [19], the total number of oracles neededfor approximating the evolution operator e − iG X t is of or-der O ( | t | log | t | ) . This requires implementing other simu-lation “tricks” to reduce the oracle cost, such as reducingthe Hamming weight of the state of the ancillas for eachsimulation of e − isO X ⊗ D c , coming from the Trotter-Suzukiapproximation (see Ref. [19] for more details). Appendix C: The modified Hamiltonians ˜ G X The first modified Hamiltonian we analyze is the onein Eq. (12) for Grover’s algorithm, and write ˜ G X = ˜ H U X .Then, ˜ G X == − O X ⊗ X l : U l = O X [ | l ih l − | c + | l − ih l | c ] ⊗⊗ [ | l i h | a + | i h l | a ] + . . . = O X ⊗ ˜ P c − a + ˜ H s − c − a . H s − c − a is a Hamiltonian that contains terms of the sys-tem, clock, and ancilla a not included in the first term.It does not contain any term that depends on O X , i.e., itcontains only those with R (and for the modified algo-rithm). Because the set { l : U l = O X } involves only oddor even values of l (i.e., R and O X alternate in Grover’salgorithm), the operator P c − a is a sum of commutingterms, each of the form − [ | l ih l − | c + | l − ih l | c ] ⊗ [ | l i h | a + | i h l | a ] . The eigenvalues of each of these terms are ± , implyingthat k P c − a k = 1 . [1] A. Messiah, Quantum Mechanics (Dover Publications,1999).[2] S. Jansen, M. Ruskai, and R. Seiler, J. of Math. Phys. , 102111 (2007).[3] A. Ambainis and O. Regev, quant-ph/0411152 (2004).[4] D. A. Lidar, A. T. Rezakhani, and A. Hamma, J. Math.Phys. , 102106 (2009).[5] S. Boixo and R. D. Somma, Phys. Rev. A , 032308(2010).[6] A. B. Finnila, M. A. Gomez, C. Sebenik, C. Stenson, andJ. D. Doll, Chem. Phys. Lett. , 343 (1994).[7] T. Kadowaki and H. Nishimori, Phys. Rev. E , 5355(1998).[8] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser,quant-ph/0001106 (2000).[9] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lund-gren, and D. Preda, Science , 472 (2001).[10] R. Somma, C. Batista, and G. Ortiz, Phys. Rev. Lett. , 030603 (2007).[11] R. Somma and G. Ortiz, Lect. Notes Phys. , 1 (2010).[12] R. D. Somma, S. Boixo, H. Barnum, and E. Knill, Phys.Rev. Lett. , 130504 (2008).[13] S. Sachdev, Quantum Phase Transitions (CambridgeUniversity Press, UK, 2001).[14] S. P. Jordan, E. Farhi, and P. W. Shor, Phys. Rev. A ,052322 (2006).[15] D. A. Lidar, Phys. Rev. Lett. , 160506 (2008).[16] A. Mizel (2010), arXiv:1002.0846.[17] D. Berry, G. Ahokas, R. Cleve, and B. Sanders, Comm.Math. Phys. , 359 (2007).[18] N. Wiebe, D. Berry, P. Hoyer, and B. C. Sanders, J. Phys.A: Math. Theor. , 065203 (2010).[19] R. Cleve, D. Gottesman, M. Mosca, R. Somma, andD. Yonge-Mallo, Proceedings of the 41st Annual IEEESymp. on Theory of Computing pp. 409–416 (2009).[20] A. Childs and R. Kothari, Theory of Quantum Compu-tation, Communication, and Cryptography p. 94 (2011).[21] D. Aharonov, W. van Dam, J. Kempe, Z. Landau,S. Lloyd, and O. Regev, SIAM J. Comp. , 166 (2007).[22] A. Mizel, D. A. Lidar, and M. Mitchell, Phys. Rev. Lett. , 070502 (2007).[23] D. Aharonov, D. Gottesman, S. Irani, and J. Kempe,Comm. Math. Phys. , 41 (2009).[24] R. P. Feynman, International Journal of TheoreticalPhysics , 467 (1982).[25] P. Deift, M. B. Ruskai, and W. Spitzer, Quantum Infor-mation Processing , 121 (2007).[26] D. Nagaj, Local Hamiltonians in Quantum Computa-tion (PhD Thesis, Massachusetts Institute of Technology,2008).[27] S. Lloyd, quant-ph/0805.2757 (2008).[28] A. J. Landahl, M. Christandl, N. Datta, and A. Ek-ert, Proceedings of the 7th International Conference onQuantum Communication, Measurement and Computingpp. 215–218 (2004).[29] L. K. Grover, Proceedings of the 28th Annual ACM Sym-posium on the Theory of Computing pp. 212–219 (1996).[30] C. Bennett, E. Bernstein, G. Brassard, and U. Vazirani,SIAM J. Comput. , 1510 (1997).[31] A. M. Childs, E. Deotto, E. Farhi, J. Goldstone, S. Gut-mann, and A. J. Landahl, Phys. Rev. A , 032314(2002).[32] A. Y. Kitaev, arxiv:quant-ph/9511026 (1995).[33] S. Boixo, E. Knill, and R. D. Somma, Quantum Inf. andComp. , 833 (2009).[34] S. Bravyi and B. Terhal, SIAM J. Comput. , 1462(2009).[35] R. D. Somma and S. Boixo, SIAM J. Comp , 593(2013).[36] A. Feiguin, R. D. Somma, and C. D. Batista (2013),arXiv:1303.0305.[37] To be rigorous, the state | ν i has to be redefined as | ν i ⊗| i a for this case.[38] While the subspace of eigenvalue zero of ˜ G X is highlydegenerate, the degeneracy is irrelevant and can be easilyremoved by adding other X -independent terms [35].[39] The frustration free property can be obtained by adding,for example, a constant to H1l