Solving strongly correlated electron models on a quantum computer

Dave Wecker,1 Matthew B. Hastings,2, 1 Nathan Wiebe,1 Bryan K. Clark,2, 3 Chetan Nayak,2 and Matthias Troyer4 1Quantum Architectures and Computation Group, Microsoft Research, Redmond, WA 98052, USA 2Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA 3UIUC 4Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland One of the main applications of future quantum computers will be the simulation of quantum mod- els. While the evolution of a quantum state under a Hamiltonian is straightforward (if sometimes expensive), using quantum computers to determine the ground state phase diagram of a quantum model and the properties of its phases is more involved. Using the Hubbard model as a prototypical example, we here show all the steps necessary to determine its phase diagram and ground state properties on a quantum computer. In particular, we discuss strategies for efficiently determining and preparing the ground state of the Hubbard model starting from various mean-field states with broken symmetry. We present an efficient procedure to prepare arbitrary Slater determinants as initial states and present the complete set of quantum circuits needed to evolve from these to the ground state of the Hubbard model. We show that, using efficient nesting of the various terms each time step in the evolution can be performed with just O(N) gates and O(log N) circuit depth. We give explicit circuits to measure arbitrary local observables and static and dynamic correlation functions, both in the time and frequency domain. We further present efficient non-destructive ap- proaches to measurement that avoid the need to re-prepare the ground state after each measurement and that quadratically reduce the measurement error.

I. INTRODUCTION or spins are mapped most naturally to the states of a general-purpose quantum computer, fermionic systems can also be simulated by using a Jordan-Wigner trans- Feynman envisioned that quantum computers would formation to represent them as spin systems, with a cost enable the simulation of quantum systems: an initial of log N, where N is the system size .5,6 In order to evolve quantum state can be unitarily evolved with a quantum the system on a general purpose quantum computer, we computer using resources that are polynomial in the size decompose the time-evolution operator according to the of the system and the evolution time.1 However, it is not Trotter-Suzuki decomposition7,8; the number of time in- clear that this would enable us to determine the answers tervals in such a decomposition is determined by the de- to the questions of greatest interest to condensed mat- sired accuracy. The evolution for each time interval is ter physicists. The properties most readily measured in expressed in terms of the available gates. If the Hamil- experiments do not appear to be simple to determine in tonian can be broken into k non-commuting terms, then a quantum simulation, and known algorithms for quan- the gates can be broken into k sets; within each set, the tum simulation do not return the quantities of great- gates can be applied in parallel, and only O(k) time steps est physical interest. In experiments in a solid, one can are needed for each interval. Time evolution can be used rarely, if ever, prepare a known simple initial state, nor in conjunction with the quantum phase estimation2–4 al- does one know precisely the Hamiltonian with which it gorithm to find an approximate energy eigenvalue and would evolve. Meanwhile, the ground state energy (or eigenstate of the Hamiltonian. the energy of any eigenstate yielded by quantum phase The challenge when applying this approach to elec- estimation2–4) is not a particularly enlightening quan- tronic systems is that the Coulomb Hamiltonian in sec- tity and, furthermore, given a quantum state it is not ond quantized form clear how to determine its long-wavelength ‘universal’ properties. Consider, for instance, the following ques-

arXiv:1506.05135v1 [quant-ph] 16 Jun 2015 N N X 1 X tions: given a model Hamiltonian for the electrons in a H = h c† c + h c† c†c c . (1) pq p q 2 pqrs p q r s solid, such as the Hubbard model, can we determine if p,q=1 p,q,r,s=1 its ground state is superconducting? Can we determine why it is superconducting? In this paper, we show that it has k = O(N 4) terms for N orbitals. Here the oper- is possible, in principle, to answer the first question and, † ators cp and cp create and annihiliate an electron with if it has a clear answer (in a sense to be discussed), the spin σ in spin-orbital p (combining orbital index i and second question as well. Moreover, we show that it is not spin index σ). The quadratic hpq-terms arise from the only possible in principle but, in fact, feasible to answer kinetic energy of the electrons and the Coulomb poten- such questions with a quantum computer of moderate tial due to the nuclei (which are assumed to be classi- size. cal in the usual Born-Oppenheimer approximation) and The first step in the solution of a quantum Hamilto- the quartic hpqrs-terms encode the Coulomb repulsion. nian is to map its Hilbert space to the states of a quantum In order to estimate the ground state energy of a sys- computer. While the Hilbert spaces of systems of bosons tem of interacting electrons in a large molecule of N or- 2 bitals, naive estimates indicated that O(N 8) operations to nearest-neighbor hopping and on-site interactions. Of are needed9, due to the large number of non-commuting course, a real solid will have long-ranged Coulomb in- terms in the Hamiltonian, but more recent analyses indi- teractions and longer-ranged hopping terms. However, cate that ∼ N 5 operations may be sufficient.10,11 While the point of a model such as the Hubbard model is not this makes the simulation of small classically-intractable to give a quantitatively accurate description of a solid, molecules feasible, the scaling is still too demanding to but rather to capture some essential features of strongly- perform electronic structure calculations for more than a correlated electrons in transition-metal oxides. For these few hundred orbitals. This makes the brute-force simula- purposes, a simplified model that does not include all tion of large molecules and crystalline solids impractical. of the complexity of a real solid should be sufficient (al- One approach to reduce the scaling complexity for the though, for the case of the cuprate high-Tc superconduc- simulation of crystalline solids is to focus on effective tors, the required simplified model may be a bit more models that capture only the relevant bands close to complicated than the Hubbard model; we focus here on the Fermi energy, and simplify the O(N 4) terms of the the Hubbard model for illustrative purposes). Coulomb interaction to just a few dominant terms. One A central question in the study of a model of strongly of the paradigmatic effective models for strongly corre- correlated electrons is: what is its phase diagram as a lated fermions, discussed in detail in Sec.II, is the square function of the parameters in the model? The Hubbard lattice Hubbard model, which is a radical simplification of model has the following interesting subquestion: is there the full electronic structure Hamiltonian (1) to arguably a superconducting phase somewhere in the phase dia- the simplest interacting single-band model, described by gram? Once the phase diagram has been determined, we the Hamiltonian wish to characterize the phases in it by computing key X X  † †  quantities such as the quasiparticle energy gap and phase H = − t c c + c c Hub ij i,σ j,σ j,σ i,σ stiffness (superfluid density) in a superconducting phase. σ hi,ji To answer these questions we will need to determine the X X +U ni,↑ni,↓ + ini, (2) ground state wave function for a range of interaction pa- i i rameters and densities and then measure ground state † correlation functions. where n = c c is the local spin density and n = i,σ i,σ i,σ i We take the following approach to solve the Hubbard P n is the total local density. Of all the orbitals in σ i,σ model (or related models) on a quantum computer: a unit cell, we only keep a single orbital (describing e.g. the Cu dx2−y2 in a cuprate superconductor), reduce the 1. Adiabatically prepare an approximate ground state long-range Coulomb repulsion to just the local repulsion |Ψ˜ 0i of the Hubbard model starting from different U within this orbital, and the hybridizations tij = −hpq initial states. to nearest neighbors on the lattice. The on-site ener- 2. Perform a quantum phase estimation to project the gies will be called i = hii. Usually one can consider a translationally-invariant model and then drop the indices true ground state wave function |Ψ0i from the ap- proximate state |Ψ˜ i, and measure the ground state in tij and i since all sites are equivalent. Here we will 0 keep them explicitly since they may not all be the same energy E0 = hΨ0|H|Ψ0i. during the adiabatic evolution. 3. Measure local observables and static and dynamic The computational complexity of the Hubbard model correlation functions of interest (2) is much reduced compared to the full electronic struc- ture problem (1), since only a single orbital is used per The general strategy for simulating quantum mod- unit cell instead of dozens needed for the full problem els has already been discussed previously, starting from and the number of terms is linear in N instead of scaling Feynman’s original proposal.1 Abrams and Lloyd12 were with the fourth power N 4. On a lattice of 20 × 20 unit the first to discuss time evolution under the Hubbard cells this means a reduction of the total number of terms Hamiltonian, but without giving explicit quantum cir- from about ∼ 1012 to ∼ 103, which makes such a sim- cuits; in Ref. 13 the same authors discuss how quantum ulation feasible on a quantum computer. Furthermore, phase estimation can be used to project from a trial state since the 2D Hubbard Hamiltonian can be expressed as into an energy eigenstate. Refs.6 and 14 provided more a sum of five non-commuting terms (the on-site interac- details of how to simulate quantum lattice models, pro- tion terms and four sets of hopping terms, one for each posed an algorithm to construct arbitrary Slater determi- nearest neighbor to a given site) each time step in the nants and discussed measurements of various observables Trotter-Suzuki decomposition requires only ∼ log N par- of interest. allel circuit depth (in fact, constant, if it were not for the In this paper we go beyond the earlier work in sev- Jordan-Wigner strings). To achieve this optimal scaling eral crucial aspects. Besides presenting full details of the we use a Jordan-Wigner transform combined with opti- necessary quantum circuits for time evolution (Sec.IV) mal term ordering and the “nesting” and Jordan-Wigner and measurements (Secs.VI andVII) we address crucial cancellation technique of Ref. 10. issues that have not received much attention so far. In The simplifying features of the Hubbard model that af- particular, in Sec.III we discuss in detail how to adia- ford such advantageous scaling with N are the restriction batically prepare approximate ground states |Ψ˜ 0i of the 3

Hubbard model starting from different initial states, cap- plification of the full Hamiltonian of a solid. A complete turing various proposed broken symmetries, and use this description of a solid takes the form: to determine the phase diagram of the Hubbard model. 2 2 2 Having thereby deduced the correct initial state, we can X pi X Pa X e cheaply prepare multiple copies of the ground state of the H = + + 2me 2Ma |ri − rj| system. i a i>j 2 2 Due to the benign scaling of circuit depth with log N, X Zae X ZaZbe + + (3) the ground state of the Hubbard model on large lattices |ri − Ra| |Ra − Rb| can be prepared in very short time of less than a second, i,a a>b even assuming logical gate times in the µs range as we where i, j = 1,...,Ne and a, b = 1,...Nions; Ne and discuss in Sec.V. Nions are, respectively, the number of electrons and ions; We then present, in Sec.VI, details of how to measure and Za is the atomic number of ion a and Ma its mass. In various quantities of interest, including both equal time certain situations, the physics is dominated by electron- and dynamic correlation functions. For the latter, we electron interactions. The ions form a crystalline lattice present a new approach, based on importance sampling in and the vibrations of this lattice (phonons), though quan- the frequency domain to complement the usual approach titatively important, do not, it is hypothesized, play a of measuring in the time domain. major qualitative role. Then we may, in a first approx- Since each measurement only gives limited informa- imation, ignore the dynamics of the ions and, thereby, tion, the ground state has to be prepared many times, obtain a purely electronic Hamiltonian which, in general, which leads to a significant total runtime of a hypothet- will take the form of Eq. (1). Explicitly introducing la- ical quantum computer. In Sec.VII, we present new bels for spin, unit cells and orbitals with a unit cell of a approaches to non-destructive measurements of arbitrary periodic solid we write this Hamiltonian as observables that substantially reduce the runtime. By us- ing a relatively cheap quantum phase estimation to recre- X   X H = − tab c† c + h.c. +  n , ate the ground state after a measurement, we reduce the ij i,a,σ j,b,σ a i,a number of times the ground state needs to be prepared i,j;a,b;σ i,a −1 X from scratch. Furthermore their runtime scales as O( ) + Uani,a(ni,a − 1)/2 −2 with the desired accuracy  instead of the O( ) scaling i,a of the na¨ıve approach to measurements. 0 X abcd † † For the aficionado of quantum algorithms, the paper + Vijkl ci,acj,bck,ccl,d + ... (4) contains the following new algorithmic ideas. In Sec.V, i,j,k,l;a,b,c,d we introduce a variant of quantum phase estimation that gives a two-fold reduction in the number of time steps and Here, i, j label unit cells and a, b label orbitals within a unit cell. The local density in orbital a is ni,a = a four-fold reduction in rotation gates compared to the P standard approach. SectionIII presents an approach to σ ni,a,σ. The coupling constants have the following a,b speeding up adiabatic state preparation by adiabatically meanings: ti,j is the matrix element for an electron to changing the Trotter time step. In Sec.VI, we present hop from orbital b in unit cell j to orbital a in unit cell an algorithm for measuring dynamical correlation func- i (possibly two different orbitals within the same unit tions by importance sampling of the spectral function in cell); a is the energy of orbital a in isolation; Ua is the the frequency domain. Finally, Sec.VII presents two energy penalty for two electrons to occupy the same or- approaches to non-destructive measurements. bital in the same unit cell. This term can be rewrit- P ten equivalently as i,a Uani,a,↑ni,a,↓. The prime on the summation in the final line indicates that we have omit- II. THE HUBBARD MODEL ted the term with i = j = k = l and a = b = c = d, which has been separately included as the Ua term. They A. Derivation of the model include, for instance, the terms with i = k 6= j = l, a = c 6= b = d, which is the density-density interaction V ababn n , and the terms with j = l, b = d, which The square lattice Hubbard model is one of the ijij i,a j,b abcb † paradigmatic models for strongly interacting fermions. is correlated hopping term Vijkj ci,ack,cnj,b. These terms a,b Although it was originally introduced as a model for will be assumed to be much smaller than ti,j , Ua and will itinerant-electron ferromagnetism15,16, it is now studied be dropped. primarily as a model for the Mott transition and broken- If the differences between the as are the largest energy symmetry (and, perhaps, even topological) ordering phe- scales in the problem, then there will be, at most, a single nomena in the vicinity of this transition (for recent re- partially-filled orbital and we can ignore all of the other views, see, for instance, Refs. 17 and 18). In particular, orbitals when studying the physics of the model at tem- it has been hypothesized that the square lattice Hubbard peratures less than |a −b|. These energy differences can model captures the important ingredients of the cuprate be on the order of eV and, therefore, such an approxima- high temperature superconductors. It is a radical sim- tion will be valid over a very large range of temperatures 4

ab and energies. Similarly, if tii is a large energy scale, it derive an effective Hamiltonian at energies much lower may be possible to reduce the model to one with a single than U by taking into account these virtual hopping pro- orbital per unit cell (which, in this case, will be a linear cesses. It takes the form: combination of the orbitals in Eq. (4)). If the on-site X H = J S · S + O(t3/U 2) (5) interactions U in this orbital (we drop the subscript a i j since there is only a single orbital under consideration) hi,ji are much larger than the interactions with nearby unit where J = 4t2/U. The O(t3/U 2) terms are due to vir- cells (nearest-neighbor, next-nearest neighbor, etc.), then tual processes in which multiple hops occcur .21 If these we can drop the latter. Similarly, if the nearest-neighbor terms are neglected, this is the antiferromagnetic Heisen- hopping is much larger than more distant hopping matrix berg model. Its ground state has an antiferromagnetic elements, then we can drop the latter and write down the moment of order 1. The temperature scale for the onset simplified model of Eq. (2). This is the Hubbard model. of antiferromagnetic correlations is ∼ J, which is a much In the case of the copper-oxide superconductors, the lower scale than the charge gap ∼ U, unlike in the small- 22 only orbital that is retained is the Cu dx2−y2 orbital. U limit, where both scales are comparable. When the (However, it has also been argued that a 3-band model subleading O(t3/U 2) terms are kept, the system remains including O px and py orbitals is necessary to capture insulating, but the spin physics may change considerably. the physics of the cuprates.19) Usually one can drop the The phase diagram of such spin models is the subject of indices in tij, Ui and i since all sites are equivalent, but considerable current research (see, for instance, Refs. 23 we will keep them here since during the adiabatic solution and 24). they will not all be the same. The situation is even less clear when the system is Due to the presumed separation of energy scales in (4), “doped”, i.e. when the density is changed from half- the effective model (2) is much simpler than (4) and has a filling. Then, an effective model can be derived to lowest- much smaller Hilbert space. As we will discuss in detail in order in t/U. The model is governed by the t − J Hamil- this paper, this simplification makes simulation of 20×20 tonian: or more unit cells feasible on a quantum computer. X X  †  X 3 2 H = − tij ci,σcj,σ + h.c. +J Si·Sj+O(t /U ) hi,ji σ hi,ji (6) B. The physics of the Hubbard model together with a no-double-occupancy constraint. As a result of this constraint, the t − J model has a smaller At half-filling (one electron per unit cell) the Hub- per site Hilbert space than the Hubbard model, which bard model gives a simple account of Mott insulating makes it a much more attractive target for exact diago- behavior.20 For U = 0, the ground state is metallic. nalization. However, the t − J model does not capture There is a single band and it is half-filled; the Fermi all of the physics of the Hubbard model (especially in surface is the diamond |kx ± ky| = π/a, where a is the the regime in which t/U is small but not so small that lattice constant. However, for U > 0, the ground state is higher-order in t/U terms can be neglected). insulating. While this is true for all U > 0, the physics Any deviation from half-filling will cause the conduc- is different in the small U  t and U  t limits. For tivity to be non-zero in a completely clean system, in U  t, the system lowers its energy by developing an- which i is independent of i in Eq. (2). However, in any tiferromagnetic order, as a result of which the unit cell real material, there will be some disorder due to impuri- doubles in size. There are now two electrons per unit ties, and the system will remain insulating for densities cell and the system is effectively a band insulator. The very close to half-filling. As the density deviates from antiferromagnetic moment N and the charge gap ∆charge half-filling, antiferromagnetic order is suppressed and −ct/U are both exponentially-small N ∼ ∆charge ∼ t e for eventually disappears. If the Hubbard model captures some constant c. For large U, i.e. U  t, on the other the physics of the cuprate superconductors, then super- 25 hand ∆charge ∼ U. In this limit, it is instructive to ex- conducting order with dx2−y2 pairing symmetry must pand about t = 0. Then, the ground state is necessarily occur for a density of 1 − x electrons per unit cell, with < < insulating: there is one electron on each site and an en- x in the range 0.05 ∼ x ∼ 0.25. Moreover, on the under- < < ergy penalty of U to move any electron since some site doped side of the phase diagram, 0.05 ∼ x ∼ 0.15, other would have to be doubly-occupied. For small but non- symmetry-breaking order parameters, such as stripe or- zero t/U, this picture is dressed by small fluctuations in der, d-density wave order, or antiferromagnetic order may which an electron makes virtual hops to a neighboring also develop (for recent reviews, see, for instance, Refs. site and then returns. As a result of these fluctuations, 17 and 18; see, also, Ref. 26 for a recent calculation find- there is an effective interaction between the spins of the ing several competing orders). On the other hand, if the electrons on neighboring sites. Since an electron can only Hubbard model is not superconducting in this doping hop to a neighboring site if its spin forms a singlet with range, then superconductivity in the cuprates must be the spin of the electron there (due to the Pauli princi- due to effects that are missing in the Hubbard model, ple), the energy is lowered by such processes if neighbor- such as longer-ranged interactions, longer-ranged hop- ing spins are antiferromagnetically-correlated. We can ping, interlayer coupling, and phonons. 5

Thus, as far as the cuprates are concerned, the prin- than exponentially, thereby allowing the evolution to oc- ciple aim of an analysis of the Hubbard model is to de- cur in polynomial time. In our protocol for solving the termine if it has a dx2−y2 superconducting ground state Hubbard model, however, any gap closing, either poly- < < for U/t ∼ 8 over the range of dopings 0.05 ∼ x ∼ 0.25. nomial or exponential – corresponding, respectively, to a If the answer is in the affirmative, then various proper- second- or first-order phase transition – is an invitation ties of this superconducting ground state can be studied. to repeat the procedure with a different initial state until For instance, the gap and its dependence on the doping, no gap closing occurs. When the initial state is in the ∆(x), can be compared to experimental measurements. same phase as the ground state of the Hubbard model, If the Hubbard model does not superconduct, then dif- the adiabatic evolution can be done in constant time up ferent models must be considered, restoring terms that to subpolynomial factors (seeVC for a more detailed dis- have been dropped in the passage to the Hubbard model. cussion of errors in adiabatic evolution; while in most of the paper we treat the time required for adiabatic evolu- tion as scaling with inverse gap squared, in that section III. ADIABATIC PREPARATION OF THE we provide a more precise discussion of diabatic errors GROUND STATE (i.e., those transitions out of the ground state that are due to a finite time scale for the evolution) and further show how it is possible to improve the scaling with gap We start the adiabatic preparation from the ground by using smoother annealing paths. state of some Hamiltonian whose solution is known. For the sake of concreteness, we consider two examples. The first, in Sec.IIIA, is the quadratic mean-field Hamilto- nian whose exact ground state is the BCS ground state A. Adiabatic Evolution from Mean-Field States for a d-wave superconductor or the analogous Hamil- tonian for any other ordered state such as striped27, 1. Mean-Field Hamiltonians antiferromagnetic28, d-density wave29, etc. states. The second, discussed in Sec.IIIB is the Hubbard model Let us suppose that we wish to test whether the ground with hopping matrix elements tuned so that the system state of the 2D Hubbard model is superconducting for breaks into disconnected 2 × 2 plaquettes. In either case, some values of t, U and density. We can do this by seeing the ground state is known and there is a gap to all excited if the Hubbard model is adiabatically connected to some states. simple superconducting Hamiltonian. It need not be real- An essential point is that these initial states can be istic; it merely needs to be a representative Hamiltonian efficiently prepared on a quantum computer. Our ap- for a superconductor. We can take the BCS mean-field proach to preparing a Slater determinant, discussed in Hamiltonian: Sec. IIIA2, is both deterministic and has better scaling MF X X  † †  than the algorithm previously proposed in Ref.6. HDSC = − tij ci,σcj,σ + cj,σci,σ We then adiabatically evolve the Hamiltonian into the hi,ji σ X 2 2   Hamiltonian of the Hubbard model with an additional − ∆x −y c† c† − c† c† + h.c., (7) weak symmetry-breaking term (to give a gap to Gold- ij i↑ j↓ i↓ j↑ stone modes, a subtlety that is discussed in SectionIIIA). hi,ji In Sec.IV we review the quantum circuit used to im- 2 2 2 2 where ∆x −y = ∆/2 for i = j ± xˆ and ∆x −y = −∆/2 plement time evolution of the Hubbard model and show ij ij for i = j ±yˆ. This is a mean-field d 2 2 superconductor how it can be done with O(N) gates and a parallel circuit x −y (DSC) with fixed, non-dynamical superconducting gap of depth of only O(log N) operations per time step. magnitude ∆. This step is formally similar to adiabatic quantum The ground state of this Hamiltonian can be written in optimization30, but with a different viewpoint. If no the form of a Slater determinant by making a staggered phase transition occurs along this adiabatic evolution, † then we conclude that we guessed correctly about the particle-hole transformation on the down-spins: ci↓ → i phase of the system (for this value of parameters in the (−1) ci↓. Then the Hamiltonian takes the form: final Hubbard model). However, if a phase transition X X   does occur, then the gap will close and we will know that H˜ MF = − t c† c + c† c we guessed incorrectly and can repeat the procedure with DSC ij i,σ j,σ j,σ i,σ σ a different initial state in a different phase. Through a hi,ji X j x2−y2  † †  process of elimination, we determine the phase of the − (−1) ∆ij ci↑cj↓ − cj↑ci↓ + h.c., (8) system for a given set of parameters and, by repeat- hi,ji ing this process for different Hubbard model parameters, we map out the phase diagram. In adiabatic quantum Under this transformation, the number of up-spin elec- 30 optimzation , it is assumed that a phase transition oc- trons per unit cell is particle-hole transformed: ni↓ → curs and it is hoped that the minimum energy at the 1 − ni↓ while the number of up-spin electrons is un- transition point scales polynomially in system size rather changed; equivalently, the chemical potential for up-spin 6 electrons is flipped in sign, µ↓ = −µ↑. Consequently, More concretely, we adiabatically evolve the Hamilto- the number of down-spin electrons is not equal to the nian according to: number of up-spin electrons. In addition, the supercon- x2−y2 MF ducting pair field ∆ij has become transformed into a HDSC (s) = (1 − s)HDSC + sHHub staggered magnetic field in the S direction with a d 2 2 X 2 2   x x −y − w∆x −y c† c† − c† c† + h.c. form factor (a d-wave spin-density wave, in the terminol- ij i↑ j↓ i↓ j↑ ogy of Ref. 29). Thus, the ground state of this quadratic hi,ji Hamiltonian is a Slater determinant of the form: X xy  † † † †  − i∆jk cj↑ck↓ − cj↓ck↑ + h.c. (11) Y † † hhj,kii |Ψi = (ukck,↑ + vkck+Q,↓)|0i (9) k Here, w is a small dimensionless parameter that deter- Here, the momenta k range over the Brillouin zone, mines the magnitude of a small pair field that remains π π − a ≤ kx,y ≤ a , where a is the lattice constant, and throughout the adiabatic evolution. At s = 0, it simply the functions uk, vk are given by: increases the superconducting gap by a factor of 1 + w. ! At s = 1, it applies a small pair field ∝ w to the Hub- 1  u2 ≡ 1 − k bard model. If the Hubbard model does not have a su- k 2 p 2 2 k + |∆k| perconducting ground state, then w 6= 0 will induce a ! small superconducting gap ∝ w. However, if the Hub- 1 k v2 ≡ 1 + (10) bard model does have a superconducting ground state, k 2 p2 + |∆ |2 k k then w will pin the phase of the superconducting gap, where k ≡ −2t(cos kxa+cos kya) and ∆k ≡ ∆(cos kxa− which would otherwise fluctuate. The magnitude of the cos kya). In Sec. IIIA2, we show how to prepare this gap will be essentially independent of w for small w. If ground state. For now, we will simply assume that this the Hubbard model is superconducting, then the system ground state can be prepared efficiently. In computing will remain superconducting all the way from s = 0 to the properties of this state, it is important to keep in s = 1 and the gap will remain open. As long as s is mind that physical operators have been transformed ac- varied slowly compared to the minimum gap – which, † i crucially, is independent of system size – the system will cording to ci↓ → (−1) ci↓ and to use the appropriate transformed operators. remain in the ground state. The final line of Eq. (11) is a small imaginary second-neighbor superconducting gap. We now adiabatically deform this Hamiltonian into a xy xy weakly-perturbed Hubbard model. There are two rea- Here, ∆jk = u∆/2 for j = k ± (xˆ + yˆ), ∆jk = u∆/2 for sons why our final Hamiltonian is a weakly-perturbed j = k ± (xˆ − yˆ), and u is a small dimensionless number Hubbard model, rather than the Hubbard model itself: that is independent of system-size. Such a term opens a (1) in the absence of long-ranged Coulomb interactions, small gap ∝ u at the nodes (assuming that the Hubbard (which are neglected in the Hubbard model) a super- model does not have a superconducting ground state that conducting ground state will have gapless Goldstone ex- spontaneously breaks time-reversal symmetry). We take citations, and (2) a dx2−y2 superconductor has gapless u∆  xJ so that this term opens a gap at the nodes fermionic excitations at the nodes of the superconduct- but is otherwise too small to affect the development of ing gap. Although these gapless excitations are impor- superconductivity away from the nodes. tant for the phenomenology of superconductors (e.g. the In summary, Eq. (11) is a path through parameter temperature dependence of the superfluid density and the space from a simple superconducting Hamiltonian with structure of vortices), they are a nuisance in the present Slater determinant ground state to a Hubbard model that context. Hence, we weakly perturb the Hubbard model in has been perturbed by small terms that give a gap to or- order to give energy gaps to Goldstone modes and nodal der parameter fluctuations and nodal excitations but oth- fermions. However, this can be done weakly so that we erwise do not affect the ground state, as may be checked do not bias the tendency towards superconductivity (or by decreasing the magnitude of these perturbations. lack thereof). If the Hubbard model superconducts, then A similar strategy can be used to test whether the Hub- the scale of the dx2−y2 superconducting gap should be bard model has any other ordered ground state, such as ∼ xJ, where hnii = 1 − x is the average occupancy of a antiferromagnetic order. In this case, our starting Hamil- site and J = 4t2/U is the superexchange interaction. (On tonian is, instead: the other hand, if the Hubbard model does not supercon- MF X X  † †  duct, then the superconducting gap in the cuprates must HAF = − tij ci,σcj,σ + cj,σci,σ be determined by some other energy scale.) Hence, we hi,ji σ apply a fixed d 2 2 superconducting gap that is much X  † †  x −y − (−1)iN c c − c c , (12) smaller than xJ in order to pin the phase of any su- i↑ i↑ i↓ i↓ perconducting order that may develop. In addition, we i apply a small imaginary dxy superconducting gap to give We then evolve along a path analogous to Eq. (11), a small gap to the nodal fermions. Neither gap scales but with the symmetry-breaking terms in the last two with system size. lines replaced by a weak staggered magnetic field. If the 7

Hubbard model has an antiferromagnetic ground state a matrix P such that (for some values of t, U, x) then the gap will not close along this path. However, if the actual ground state of ρ = PP †. (14) the Hubbard model is superconducting for these values of the couplings, then the gap will become O(1/N z/2) Then, using this matrix P in the definition of the state where z is the dynamical critical exponent of the transi- gives the state ΨSD(ρ); note that this definition fixes tion (i.e. the gap will close, up to finite-size effects) at ΨSD(ρ) up to a complex phase. The algorithm in Ref.6 π † the point along the evolution at which superconductivity proceeds by noting that Um = exp(i 2 (bm + bm)), acting † develops. There will be a non-zero staggered magnetiza- on the vacuum, produces the state bm|0i, up to a phase. † tion throughout since we never fully remove the staggered Therefore, if the bj are suitably orthogonalized, succes- magnetic field, but there will be a sharp signature at the sively acting with the unitary Um produces the desired point at which superconductivity develops. The presence state. However, no explicit method for implementing this of a staggered field, which is kept small, may slightly shift unitary Um is given; if the unitary is implemented by a the onset of superconductivity but will have no other ef- Trotter method, this will be extremely inefficient. We fect. explain a simpler technique. This strategy can allow us, in some of the circum- Let us first assume that ρ is real; the extension to stances in which it has a clear answer, to address the the complex Hermitian case will be straightforward. The question: why does the 2D Hubbard model have a su- idea is first to prepare the Slater determinant state perconducting ground state? In particular, it is possible Ne † that the superconducting ground state of the Hubbard ΨSD(ρ0) ≡ Πj=1aj|0i. (15) model necessarily has some other type of order coexisting with superconducting order, especially in the underdoped This corresponds to the case that ρ is an n-by-n diagonal regime. These secondary orders play a role in some theo- matrix, with ones in the first Ne entries and zeroes else- ries of the cuprate superconductors. If such an order were where. This state has one electron on each of the first present in the Hubbard model, evolution from a purely Ne states. It is a product state that can be prepared in superconducting initial state would necessarily encounter linear time (indeed, simply initialize each of the first Ne a phase transition (at the point of onset of this additional spins to up and the other spins to down). Then, we act order). Therefore, we can determine not only if the Hub- on this state ΨSD(ρ) with a sequence of Givens rotations. bard model has a superconducting ground state, but also A Givens rotation, to borrow the language of matrix di- whether the ground state exhibits a secondary type of agonalization, is a rotation matrix that acts on two rows long-ranged order over some range of dopings. We will or columns at a time. For any given pair, i, j, define the return to the question “why does the Hubbard model operator Ri,j(θ) by superconduct” when we discuss correlation functions. † The virtue of using these mean field Hamiltonians Ri,j(θ) = exp(θci cj − h.c.). (16) as starting points for adiabatic evolution is that their physics is fully understood. In particular, their ground This is a simple two-qubit gate (up to Jordan-Wigner states are Slater determinants, which we can efficiently strings) and is, therefore, more straightforwardly imple- prepare, as we explain in the next section. mented than the unitary Um in the standard method described above. Define the n-by-n orthogonal ma- trix ri,j(θ) by its matrix elements as follows. Let 2. Preparing Slater Determinants (ri,j(θ))k,k = 1 for k 6= i, j and let (ri,j(θ))k,l = 0 if k 6= l and k 6= i, j or if k 6= l and l 6= i, j. Then, let (ri,j(θ))i,i = (ri,j(θ))j,j = cos(θ) while (ri,j(θ))i,j = In this section, we explain an efficient, simple quantum −(r (θ)) = sin(θ). Then, algorithm to prepare Slater determinants. This is then i,j j,i used to prepare the Hubbard ground state by adiabatic −1 Ri,j(θ)ΨSD(ρ) = ΨSD(ri,j(θ)ρri,j(θ) ), (17) evolution from U = 0 to U 6= 0. The standard algorithm is in Ref.6. This algorithm where again the equality holds up to the phase ambiguity. suffers from some drawbacks. Given a projector ρ, a Now, given some desired target ΨSD(ρ), we claim that Slater determinant state is a state one can always find a sequence of at most nNe Givens rotations that will turn the matrix ρ into the matrix ρ0. Ne † ΨSD(ρ) = Πj=1bj|0i, (13) Thus, by inverting this sequence of Givens rotations and applying that sequence to the state ΨSD(ρ0), we succeed where |0i is the no particle vacuum state, and each op- in constructing the desired state. The Givens rotations † † † erator bj is a linear combination of ai , where the ai are will be ordered such that the Jordan-Wigner strings can fermion creation operators on a given site i. We write be cancelled as in Ref. 10, so the algorithm takes time † Pn † bj = i=1 ai Pij, where Pij are complex scalars and are O(Nen). To show our claim that the sequence of Givens the entries of P , n is the number of orbitals, and Ne is rotations exists, note that the effect of a given rotation −1 the number of electrons. Given any matrix ρ, we can find which transforms ρ to ri,j(θ)ρri,j(θ) can be achived by 8 transforming P to ri,j(θ)P . Consider a left singular vec- tor of P with singular value equal to 1; we can find a sequence of at most n Givens rotations that rotate that vector to be equal to the vector (1, 0, ...). As a result, all the other singular vectors now have amplitude only on sites 2, ..., n. Find another sequence to transform some other left singular vector with singular value equal to 1 into the vector (0, 1, 0, ...). Continue until all singu- lar vectors with nonzero singular values have amplitudes only on the first Ne sites. The extension of this procedure to the complex case is simple. Rather than having the Givens rotation Ri,j(θ) depends only a single angle θ, we need to perform a rota- † tion exp(zci cj −h.c.) for some complex number z. All the time estimates remain unchanged up to constant factors. FIG. 1. Success probability as a function of total anneal- However, for many systems a much shorter sequence of ing time for an annealing path in the Hubbard model that Givens rotations can be found. Using the same idea as in begins at U = 0 and vertical hopping equal to 2 and ends the fast Fourier transform, if n is a power of 2, there exists at U = 2.5t (blue) or U = 4t (red) and all hoppings equal a sequence of n log2(n) Givens rotations to transform an to 1. The initial state is a Slater determinant state (of free n-by-n matrix to the momentum basis. Thus, we can fermion single-particle states). Annealing is linear along the produce ground states of a system of free fermions on a path. The x-axis represents the total time for the anneal. The periodic lattice in time O(n2 log(n)). This can be easily y-axis represents the probability that the phase estimation extended to handle the case of a periodic system with a routine returns an energy that is within 10−3 of the ground unit cell larger than a single site. Assume there are states state energy. This probability is obtained by sampling runs |i, ~xi where i labels an atom in a unit cell and ~x labels a of the phase estimation circuit, giving rise to some noise in the curves. lattice vector. We transform to a momentum basis |i,~ki, and then for each ~k we initialize some Slater determinant in the unit cell in momentum space. If the unit cells have sites. Then, 3 Givens rotations were used to create a size O(1), the time to initialize each cell is O(1) so the uniform superposition of the hole in various states in the 2 total time is still O(n log(n)). top row. Finally, 4 Givens rotations were used to create A further possible optimization is that if Ne > n/2, we a uniform superposition between top and bottom rows. can instead use Givens rotations to bring the holes to the This was done for both spins, giving 2×7 = 14 rotations. desired states, rather than the electrons, taking a time We then annealed to U 6= 0 while reducing the verti- O((n − Ne)n) rather than O(Nen). As an interesting cal coupling to unit strength. The results are shown in application of this, consider creating the ground state Fig.1. As seen, increasing the annealing time increases of Ne = 3 electrons in n = 4 sites, using a hopping the success probability, until it converges to 1. The an- Hamiltonian with the sites arranged on a ring. This can nealing times required to achieve substantial (say, 90%) be done by initializing a state with electrons on the first overlap with the ground state are not significantly differ- 3 sites and then apply a total of three different Ri,j(θ) ent from the joining the preparation approach discussed operators. next.

3. Numerical Results B. Preparing of d-wave resonating valence bond states from plaquettes We applied the Slater determinant preparation pro- cedure to the Hubbard model on 8 sites arranged in two 1. d-wave resonating valence bond states rows of 4 sites. We started at U = 0 and i = 0. All hori- zontal couplings were set equal to 1, while to avoid having an exact ground state degeneracy the vertical couplings An alternative approach to adiabatic state prepara- were doubled in strength to 2 (note that this requirement tion has previously been proposed in the context of ul- depends on the filling factor; for example, at half-filling tracold quantum gases in optical lattices. In this ap- we would instead prefer to remain at vertical coupling proach, d-wave resonating valence bond (RVB) states equal to 1 to avoid degeneracies). The ground state of are prepared.31 RVB states are spin liquid states of a this model is a Slater determinant of free fermion single- Mott insulator, in which the spins on nearby sites are particle wavefunctions that we could prepare with a total paired into short-range singlets. RVB states were first of 14 Givens rotations for both spin up and spin down. introduced as proposed wave functions for spin liquid This was done by initializing particles in three out of four ground states of Mott insulators32, and then conjec- sites in the top row of sites, and holes in the remaining tured to describe the insulating parent compounds of the 1 3

0 2

9

1 3 1 3 5 7

0 2

FIG. 2. Labeling of the sites in a plaquettes 0 2 4 6 1 3 5 7 cuprate superconductors.33 In more modern language, FIG. 3. Coupling of two plaquettes they are described as Z2 topologically-ordered spin liq- uid states.34 The idea was that, upon doping, the sin- glet pairs would begin to move and form a condensate of 2. ramp up the hoppings tij from 0 to t during a time Cooper pairs. Although0 the insulating2 parent4 compounds6 T1, of the cuprates are antiferromagnetically-ordered, it is nevertheless possible that the cuprate superconductors 3. ramp the potentials i down to 0 during a time T2. are close to an RVB spin liquid ground state and are −1 Time scales T1 = T2 ≈ 10t and  = U + 4t are suf- best understood as doped spin liquids. The RVB sce- ficient to achieve high fidelity. As we discuss below, nario has been confirmed for t-J and Hubbard models it can be better to prepare the state quickly and then of coupled plaquettes35,36 and ladders (consisting of two 37,38 project into the ground state through a quantum phase coupled chains) and remains a promising candidates estimation than to aim for an extremely high fidelity in for the ground state of the Hubbard model and high- 39 the adiabatic state preparation. To prepare a plaque- temperature superconductors. tte with two electrons we start from the product state Instead of starting from mean-field Hamiltonians, we † † c c |0i and use the same schedule, except that we build up the ground state wave function from small spa- 0,↑ 0,↓ choose the non-zero potentials on three sites of the pla- tial motifs, in this case four-site plaquettes, which are quette ( =  =  = ). the smallest lattice units on which electrons pair in the 1 2 3 Hubbard model.35 Following the same approach as orig- inally proposed for ultracold quantum gases, we prepare 3. Coupling of plaquettes four-site plaquettes in their ground state, each filled with either two or four electrons.31 These plaquettes then get coupled, either straight to a two-dimensional square lat- After preparing an array of decoupled plaquettes, some tice, or first to quasi-one dimensional ladders, which are of them with four electrons and some with two electrons subsequently coupled to form a square lattice. depending on the desired doping, we adiabatically turn on the coupling between the plaquettes. As a test case we use that of Ref. 31 and couple two plaquettes (see Fig.3), one of them prepared with two electrons and 2. Preparing the ground states of four-site plaquettes one with four electrons. Naively coupling plaquettes by adiabatically turning The first step is preparing the ground state of the Hub- on the intra-plaquette couplings t24 and t35 will not give bard model on 4-sites plaquettes filled with either two or the desired result, since the Hamiltonian is reflection four electrons. We start from very simple product states symmetric but an initial state with two electrons on one and adiabatically evolve them into the ground states of plaquette and four on the other breaks this symmetry. the plaquettes. No matter how slowly the anneal is done, the probability To prepare the ground states of plaquettes with of preparing the ground state will not converge to 1. We four electrons we start from a simple product state thus need to explicitly break the reflection symmetry by † † † † either first ramping up a small chemical potential on a c0,↑c0,↓c1,↑c1,↓|0i, using the labeling shown in Fig.2. Our initial Hamiltonian, of which this state is the ground subset of the sites or – more easily – not completely ramp- state has no hopping (tij = 0), but already includes the ing down the non-zero i values when preparing plaque- Hubbard repulsion on all sites. To make the state with tte ground states. Consistent with Ref. 31 we find that a −1 doubly occupied state the ground state we add large on- time T3 ≈ 50t is sufficient to prepare the ground state site potentials 2 = 3 = 2U + 3t on the empty sites. To with high fidelity. prepare the ground state of the plaquette we then See Fig.4 for success probabilities depending on times T1 to ramp up hopping in a plaquette, T2 to ramp down  1. start with tij = 0, Ui = U and a large potential in a plaquette, and T3 to join the plaquettes. The specific on the empty sites (here 2 = 3 = ) so that the couplings in these figures are an initial potential 8t on the initial state is the ground state, empty sites in the plaquette with four electrons, chemical 10

FIG. 4. Probability of preparing ground state as a function of times T , T ,T in units of t−1, following the annealing sched- 1 2 3 FIG. 6. Shown is the probability P of finding three electrons ule described above. During time T , a uniform potential  33 3 on each of the plaquette after decoupling two plaquettes with was ramped from t to 0 on the plaquette with two electrons a total of six electrons as a function of the time T (in units to break the symmetry between plaquettes. s of t−1) over which the plaquettes were nearly adiabatically separated. A decrease of P33 indicates pairing of holes in the ground state of a plaquette.

a quantum phase transition to a different phase.

4. Decoupling plaquettes

As proposed in the context of analog quantum simula- FIG. 5. Energy levels during annealing. y-axis is energy. x tion, the pairing of holes on plaquettes can be tested axis is time. From time 0 to time 16 (in units of t−1) the by adiabatically decoupling plaquettes.31 We start by hopping is ramping up, time 16 to time 36 the non-uniform  preparing one plaquette with four electrons and a second in each plaquette is ramping down (leaving a uniform  = 1 plaquette with two electrons. We then couple them as on the plaquette with two electrons), time 36 to time 76 is described above to end up with a system with two holes joining plaquettes. All levels remain non-degenerate except relative to the half filled Mott insulator. After adiabati- for a degeneracy that appears at the end of the anneal. After cally decoupling the two plaquettes again we measure the time 76 plaquettes are separated, as described below. number of electrons on each plaquette. If holes bind in the ground state, they will end up on the same plaquette, and we measure an even electron number per plaquette. potential 8t on the empty sites in the plaquette with two If they don’t bind each hole will prefer to be on a differ- electrons and chemical potential t on the occupied site in ent plaquette to maximize its kinetic energy and we will the plaquette with two electrons. See Fig.5 for a plot of end up with three electrons per plaquette. the spectral gaps for an example annealing schedule. Going from two plaquettes to the full square lattice As shown in Fig.6, the probability of finding three is straightforward: we bias some plaquettes with small electrons on a plaquette goes to zero for U = 2t and values of i to break any reflection and rotation symme- U = 4t when increasing the time Ts over which we ramp tries, and then switch on inter-plaquette hoppings, and down the inter-plaquette hopping. This indicates pairing switch off the bias potentials. The time scale here is a- and is consistent with the known critical value of U ≈ 4.5t priori unknown. According to Ref. 31 the ground state for pairing on four-site plaquettes. On the other hand for on ladders can be prepared within a relatively short time U = 6t and U = 8t we see an increase in the probability T ≈ 200t−1 − 500t−1. In two dimensions we expect that of finding three electrons per plaquette, indicating the similar time scales may be sufficient unless we encounter absence of pairing. 11

C. Trotter Errors and Accelerating adiabatic Name Symbol Matrix Representation preparation by using larger time steps " √ √ # H 1/ 2 1/ 2 Hadamard √ √ In practice, it suffices for our annealing preparation 1/ 2 −1/ 2 " √ √ # to obtain a state with sufficiently large overlap onto the Y 1/ 2 i/ 2 Y basis change √ √ ground state that phase estimation will then project onto i/ 2 1/ 2 the ground state with a reasonably large probability. We " # T (θ) 1 0 can thus tolerate some errors in our annealing path. For Phase gate this reason, we annealed using a larger time step, before 0 e−iθ " # doing the phase estimation with a smaller time step. R (θ) eiθ/2 0 Z rotation z In fact, this approach of annealing at a larger time 0 e−iθ/2 step can be turned into an approach which uses a larger   time step but still produces a state with very large over- 1 0 0 0   lap with the ground state. Note that the larger Trotter-  0 1 0 0  controlled not (CNOT)   Suzuki step means that we in fact evolve under a modified  0 0 0 1  Hamiltonian, proportional to the logarithm of the unitary 0 0 1 0 describing a single time step. As the time step decreases, this Hamiltonian becomes closer to the desired Hamil- TABLE I. An overview of the quantum gates used in this tonian. It is then possible to evolve with a larger time paper. Note that Y gates do not represent Pauli Y ; rather, step, following the adiabatic path in parameter space, they represent a Clifford operator that interchange Y and Z and then at the end of the path one can “anneal in the spin orientations. The phase gate T (θ) can be replaces by an time step”, gradually reducing the time step to zero. Rz(θ) rotation, up to a global phase that can be kept track of classically. In our simulation of the anneal using joining two pla- quettes, the system was rather insensitive to Trotter er- rors. We used a relatively large time step, equal to 0.25t−1 for the annealing, and used a much smaller time step for the phase estimation that made the errors negli- Rz(θ/2) gible there. A time step of 0.05t−1 for phase estimation yielded relative errors below 10−3 and 0.003t−1 yielded relative errors below 10−5. The larger time step used for the adiabatic preparation prevents us from obtaining 100% Rz(θ/2) Rz(−θ/2) overlap with the ground state even in the limits of very long annealing times; however, as seen we still obtain very high overlaps in the range of 80% − 96%. FIG. 7. Quantum circuit to implement time evolution for a time step θ under the term Hpqqp = np,σnq,σ0 = † † c c c 0 c . Top and bottom lines represent qubits for IV. IMPLEMENTING UNITARY TIME p,σ q,σ0 q,σ p,σ 0 EVOLUTION OF THE HUBBARD MODEL p, σ and q, σ .

A. Circuits for the individual terms

For the Hubbard model, we need to implement uni- Hpp term, is not needed for the final Hamiltonian but is tary evolution under two types of terms, hopping terms useful for annealing. The circuit for this term is shown † † † cp,σcq,σ, and repulsion terms cp,σcp,σcq,σcq,σ. These in Fig.8. Since all the repulsion terms in the Hubbard terms are a subset of those needed for earlier elec- model and all the chemical potential terms commute with tronic structure calculations. In addition we will need each other, they can be applied in parallel, thus needing to implement unitary evolution under a pairing term only O(N) gates and O(1) parallel circuit depth. † † cp,σcq,σ0 + h.c.. To establish notation we summarize In TableI the gates used in the quantum circuits shown in this paper.

1. The repulsion and chemical potential terms T (θ)

The repulsion term is an example of what is called an FIG. 8. Quantum circuit to implement time evolution for a Hpqqp term and evolution under this term is given in Fig. † † time step θ under the chemical potential term np,σ = cp,σcp,σ. 7. A chemical potential term, cp,σcp,σ, which we call and 12

H H Y Y †

† Z H Rz(θ) H Y Rz(θ) Y Z

† † FIG. 9. Quantum circuit to implement time evolution for a time step θ under the hopping term cp,σcq,σ + cq,σcp,σ. Top and bottom lines represent qubits for p, σ and q, σ.

2. The hopping and pairing terms apply a Hadamard to the first qubit and a Y basis change to the second, and in the second half we apply the Y The unitary evolution under the hopping term, that we basis change to the first qubit and the Hadamard to the second. call an Hpq term, is given by the circuit in Fig.9, and has also been considered previously. It also will be useful to be able to implement evolution under a super-conducting † † pairing term, ∆cp,σcq,σ0 +h.c.. This term is similar to the B. Optimizing the cost of the Jordan Wigner hopping term, since both are quadratic in the fermionic transformation operators. The hopping and pairing term circuits will be very similar to each other, with the changes involv- In order to implement the unitary time evolution in the ing changing some of the basis change gates. The pair- Hubbard model, we use a Jordan-Wigner transformation ing term will be needed for the adiabatic evolution from to turn the model into a spin model. Then, we use pre- the BCS mean-field Hamiltonian suggested in subsection viously proposed circuits for the problems of quantum III A 1. Also, it will be needed to measure superconduct- chemistry. The depth of the circuits needed will depend ing correlations present in the Hubbard state: in this case upon the particular ordering of spin-orbitals chosen. We the measurement will be of the correlation function be- choose to order so that all qubits corresponding to spin tween two different superconducting pairing terms on two up appear first in some order, and then all qubits corre- pairs of sites separated from each other. Measurement sponding to spin down. Since the hopping terms preserve will be discussed more later, but our general procedure spin, this simplifies the Jordan-Wigner strings needed. for measurement will require knowledge of the unitary For a given spin, we order the sites in the “snake” pat- which implements evolution under the term. tern shown in Fig. 11. For the case ∆ real, we wish to implement unitary evo- With this snake pattern, we group the hopping terms † † lution by exp[−iθ(cp,σcq,σ0 + h.c.)]. Let us write Xp,σ to into four different sets, as shown. The terms in any given denote the Pauli X operator on the qubit corresponding set all commute with each other. The two horizontal sets to spin-orbital p, σ, and similarly Yp,σ and Zp,σ denote of terms have no Jordan-Wigner string required, since Pauli Y and Z operators on that spin-orbital. Using the they involve hopping between neighboring sites with the † identity that c = (1/2)(X + iY ) and c = (1/2)(X − iY ), given ordering pattern, and so these terms can all be up to phase factor associated with the Jordan-Wigner executed in parallel. The vertical terms do not commute strings, we wish to implement with each other. However, they nest as in Ref. (10) so θ that all vertical terms in a given√ set can be executed with exp[−i (Xp,σXq,σ − Yp,σYq,σ0 )ZJW ], (18) O(N) gates using a depth O( N), where we assume that 2 the geometry is roughly√ square so the the number of sites where ZJW denotes the product of the Pauli Z operators in a given row is N. on the spin-orbitals intervening between p, σ and q, σ0. For periodic boundary conditions, we must also handle In general, we wish to implement this unitary operation the terms going from left to right boundary or from top controlled by some ancilla. In fact, this circuit is the same to bottom boundary. The terms going from left to right † as the circuit used to implement the term cp,σcq,σ0 +h.c., boundary can be executed in parallel with each other. up to a sign change on the second spin rotation, as shown Each term naively requires a depth proportional to the in Fig. 10. length of a given horizontal row to calculate the Jordan- For ∆ imaginary, we wish to implement Wigner√ string. Assuming a square geometry, this length θ is O( N). We can use a tree to calculate the fermionic exp[−i (Xp,σYq,σ + Yp,σXq,σ0 )ZJW ]. (19) parity of the sites in the middle of the string, reducing 2 the depth to log(N) if desired. For the terms from top to This circuit is similar to the other pairing and hopping bottom edge, there are Jordan-Wigner strings of length circuits, except that in the first half of the circuit, we O(N), but these terms nest and so can all be executed in 13

H H Y Y †

† Z H Rz(θ) H Y Rz(−θ) Y Z

† † FIG. 10. Quantum circuit to implement time evolution for a time step θ under the pairing term cp,σcq,σ0 + cq,σ0 cp,σ. Top and bottom lines represent qubits for p, σ and q, σ0.

phase estimation to project an approximate state onto the ground state may be preferred. The gap will depend upon the annealing path; since we use a symmetry break- ing field until nearly the end, the gap only closes at the end of the anneal. In this section we first give a simple improvement to quantum phase estimation that leads to a reduction in depth. We then give some numerical esti- mates for annealing times for free fermion systems (the simplest case where one can estimate annealing times for a large system numerically).

A. Reducing the number of rotations in quantum FIG. 11. Snake pattern for ordering of sites. Arrow shows phase estimation order of sites. Dashed and solid lines represent hopping terms. The hopping terms are grouped into four different sets, two sets of vertical and two sets of horizontal hoppings as shown While this general setting of using adiabatic prepara- by different colorations and dash patterns. tion and quantum phase estimation has been considered many times before, we briefly note one simplification of the quantum phase estimation procedure which leads to parallel, reducing the depth to O(log(N)). The fermionic at least a twofold (and, depending upon the available gate parity on the sites on the middle rows (those other than set, fourfold) reduction in the depth. This simplification the top and bottom row) can be calculated using a tree, is not in any way specific to the Hubbard model, and again in depth O(log(N)). The total depth can then be could be applied in other settings, such as in quantum √ chemistry. The usual approach to quantum phase esti- O(log(N) N). In fact, using a tree to compute parities, mation is to build controlled unitaries to implement the the total depth can be reduced to polylogarithmic in N. unitary evolution (by using the ancilla to control the uni- taries described above). In this case, if the ancilla qubit is in the |0i state, then no quantum evolution is imple- V. FINDING THE GROUND STATE BY mented, while if the ancilla qubit is in the |1i state, then QUANTUM PHASE ESTIMATION quantum evolution by U = exp(−iHt) is implemented (up to Trotter-Suzuki error). Then, quantum phase esti- The annealing algorithm generates a state with good mation estimates the eigenvalues of this unitary U. The overlap with the ground state if the rate is sufficiently simple modification that we propose is that instead, if the slow.VC discusses optimized paths that increase the ancilla qubit is in the |0i state, then we implement evo- overlap. However, in fact it is not necessary to obtain lution by exp(iHt/2), while if the ancilla qubit is in the a perfect overlap with the ground state; if the overlap |1i state, then we implement evolution by exp(−iHt/2). with the ground state is reasonably large, then we can Since the difference between these two evolutions is the apply quantum phase estimation to the state and have same exp(−iHt), then the same phase estimation proce- a reasonably large chance of projecting onto the ground dure on the ancilla will return the identical result for the state. Since the time for the simplest annealing path energy (again up to Trotter-Suzuki error). The primary scales inversely with the gap squared, while the time to advantage of this approach is that we have then only to use quantum phase estimation to measure energies more implement unitary evolution for half the time (i.e., t/2 accurately than the gap (and thus to determine whether rather than t), so this allows the depth of the circuits to or not one is in the ground state) scales inversely with be reduced by a factor of two. the gap, up to logarithmic corrections, using quantum In fact, when we examine in detail how to im- 14 plement this controlled evolution by exp(+iHt/2) or improved annealing strategies that overcome some of the exp(−iHt/2), we find that a further twofold improve- effects of the endpoints. ment will occur in many cases. In the usual approach, To put some numbers onto these generalities, we con- the controlled evolution is implemented by having the sidered a system of spinless free fermions at half-filling in arbitrary angle Z rotations replaced by controlled Z ro- one dimension with periodic boundary conditions (this tations. For example, in the circuit of Fig.9, the two of course is the most advantageous case with only two rotations by angles −θ/2 are replaced by controlled ro- modes with minimum gap). We follow the annealing path tations. A controlled rotation by angle θ implements no X  †  X  †  rotation if the ancilla is in the |0i state and implements Hs = ci ci+1 + h.c. + s ci ci+1 + h.c. , (20) a rotation by angle θ if the ancilla is in the |1i state. i even i odd If we have access to arbitrary Z rotations by angle θ, starting at a fully dimerized state at s = 0 and moving and to Clifford operations, then the controlled rotation to a uniform state at s = 1, where c† (c ) creates (an- by angle θ is implemented by rotating the target by an- i nihilates) a spinless fermion on site i . We followed a gle θ/2, then applying a controlled NOT from the ancilla uniform annealing schedule and used a time discretiza- to target, then rotating the target again by angle θ/2, tion with negligible Trotter error. We chose sizes which then again applying a controlled NOT from the ancilla were not a multiple of 4 to avoid having an exact zero to target, so that the net rotation is θ/2 − θ/2 = 0 if mode. the ancilla is in the |0i state and is θ/2 + θ/2 = θ if the For a system 1002 sites, with an annealing time t = ancilla is in the |1i state. For the simplified approach, 1000, the probability of ending in the ground state is we need instead to implement a rotation by angle −θ if 0.54... Thus, the expected time to end in the ground state the ancilla is in the |0i state and by +θ if the ancilla is roughly 2000 plus an roughly additional 2 phase esti- is in the |1i state. This can be done using a controlled mation steps on average. Reducing the annealing time NOT from ancilla to target, followed by a rotation of the to t = 500, the probability of ending in the ground state target by angle −θ, followed by another controlled NOT is 0.306..., taking slightly less expected annealing time, from ancilla to target. Thus, this uses only half as many but more phase estimation steps on average. Increasing arbitrary rotations, which are expected to be the most to 2002 sites, the situation is worse. For t = 1000, the expensive gates. probability of ending in the ground state is 0.13..., so the expected time is over 7500, plus over 7 phase estimation steps on average. Because the system is translationally B. Annealing Time for Free Fermions invariant with period 2 for all s, the probability of ending in the ground state is a product over momentum vectors To make some estimate of the cost of adiabatically of the probability of ending in the ground state in each preparing a good approximation to the ground state, it momentum vector. Each of these probabilities can be is worth considering the case of free fermions. For any calculated by evolving a 2-by-2 matrix. In all cases, the system of free fermions with translation symmetry and dominant contribution to the smallness of the probabil- k sites per unit cell, the annealing problem decouples ity to end in the ground state was from the lowest energy into many different problems, one for each Fourier mode, mode. Thus, small changes in the low energy density of each problem describing annealing of a k-dimensional states can lead to dramatic changes in the success prob- subspace. For k = 2, this is a Landau-Zener problem. ability. For a single such Landau-Zener problem, the annealing For 1002 sites, the lowest eigenvalue of the free fermion time is expected to scale as 1/∆2 where ∆ is the gap of hopping matrix is 0.0188..., so the gap ∆ of the Hamil- the Hamiltonian. However, we have many Landau-Zener tonian is twice that. Hence, in this case, the phase esti- problems in parallel, and we want all Fourier modes to mation time could be much smaller than the annealing reach the ground state. The problem is least serious if time. the Fermi surface consists just of isolated points (such Since rather large Trotter time steps of order 0.25 are as in one dimension or for a semimetal in higher dimen- sufficient to obtain reasonable overlap with the ground sions) as there are only a few Fourier modes with small state, we find that about 104 steps are sufficient to pre- gap. For systems with a Fermi surface, the density of pare the ground state. Using parallel circuit each step states at the Fermi surface is larger, meaning that there can be performed with a small circuit depth that in- are more modes with small gap. However, the probabil- creases only logarithmically with the system size. Hence ity of a diabatic transition for a Landau-Zener system is a parallel circuit depth of about one million gates is suf- exponentially small in the ratio of the gap squared to the ficient to prepare the ground state. Landau-Zener velocity. Hence, even if there are a large number of modes with gap ∆, this should require only a logarithmic slowdown in the adiabatic velocity to have a C. Improved Annealing Paths small probability of a diabatic transition. However, one must also worry about effects due to the endpoints of the There are broadly two strategies for reducing the er- evolution. See, however,VC for a detailed discussion of ror in the annealing path. The typical objective in such 15 optimizations is to find a function, f(s), that satisfies show that, for fixed T , the error grows at most polyno- f : [0, 1] 7→ [0, 1] and f(0) = 0 and f(1) = 1 such that mially with m.41 Calculus then shows that the optimal the diabatic leakage out of the groundstate caused by value of m scales logarithmically with the error tolerance R 1 R 1 iT E0(s)ds −iT H(f(s))ds Udiab := e 0 T e 0 is minimized for and hence such contributions are subpolynomial. a fixed value of T . Here we use the convention that These two strategies are often mutually exclusive when T is the time ordering operator, T is the total evo- the avoided crossing does not occur near the boundary lution time allotted and s = t/T is the dimensionless because if H(s) is analytic then H˙ (s) will have to be time, and the phase factor is included to ensure that larger away from the boundary to compensate for the fact limT →∞ Udiab = Uadiab is well defined. that it is nearly constant in the vicinity of the boundary. The first strategy is known as local adiabatic This can cause the Hamiltonian to move rapidly through interpolation.40 The idea of local adiabatic interpolation the avoided crossing, which increases the annealing time is to choose f(s) to increase rapidly when the eigenvalue required. However, the minimum gap often occurs at the gap of H(f(s)) is large and increase slowly when the gap end of the evolution when transforming from the initial is small. This strategy can substantially reduce the time Hamiltonian (such as the free Fermion model, plaque- required to achieve a fixed error tolerance, but does not tte preparation, or other tractable approximation) to the improve the scaling of the diabatic errors in the state interacting theory. This means that the two strategies preparation, which scale as O(1/T ) for Hamiltonians that are often compatible here and that boundary cancellation are sufficiently smooth. methods will often be well suited for high–accuracy state The second strategy, known as boundary cancella- preparation. Further adiabatic optimizations could also tion is often diametrically opposed to local adiabatic be achieved by altering (22) to approximate the local– optimization.41 The idea behind it is instead of mov- adiabatic path in the region away from from s = 0, 1. ing at a speed designed to minimize diabatic transitions, We are of course concerned about the evolution time you choose a path designed to maximize cancellations required to perform Uadiab ≈ Udiab since it dictates the between them. There are several strategies for exploit- resources needed to implement P0 within a fixed error. ing this, but the simplest strategy to to choose f(s) such At first glance, the use of a Trotter decomposition may k that ∂s f(s)|s=0,1 = 0 for k = 1, . . . , m. Then if f(s) is appear to be problematic for adiabaticity because the at least m + 2 times differentiable and H(f(s)) does not time–dependent Hamiltonian that describes the decom- explicitly depend on T then the error in the adiabatic position is discontinuous. Such discontinuities are not approximation is at most42 actually problematic, as can be seen by using the trian- gle inequality m+1 k∂s H(f(s))k m+2
2 max + O 1/T , (21) ff † TS ff TS† s ∆(s)m+2(s)T m+1 k(UadiabP0 Uadiab − UdiabP0 Udiab )|E0ik ≤ 2 k(U − U )|E ik + k(U − U TS )|E ik
, where ∆(s) is the gap at the given value of s. Equa- adiab diab 0 diab diab 0 (24) tion (21) therefore implies that if where U TS is a quantum circuit approximation to the R s ym(1 − y)mdy diab f(s) = 0 , (22) finite time evolution Udiab. This shows that the error R 1 m m 0 y (1 − y) dy in the state preparation is at most twice the sum of the adiabatic error and the error in simulating the adiabatic then the upper bound on the asymptotic scaling is im- evolution. m+1 proved from O(1/T ) to O(1/T ). Furthermore, if a In order to make the entire simulation error at most δ, diabatic error of δ is desired then it suffices to take it suffices to set both sources of error to δ/2. This cre- 1 ! ates an issue because the cost of simulation scales at least k∂m+1H(f(s))k m+1 T = O max s . (23) linearly with T , which in turn scales with 1/δ. There 1+ 1 1 s ∆(s) m+1 δ m+1 are a wide array of different Trotter–based formulas that can be used in the simulation but in all such cases the This implies that even in the limit of small δ the cost of Trotter number (and in turn the simulation cost) scales43 adiabatic state preparation is not necessarily prohibitive for integer k > 0 as T 1+1/2k/δ1/2k, which using (21) is because m can be increased as δ shrinks to achieve im-  −(2k+m+2)  proved scaling with the error tolerance. Furthermore, the O δ 2k(m+1) For any fixed m, the value of k that leads scaling of the error with the minimum gap becomes near– to the best scaling with δ can found by numerically opti- quadratically smaller than would be otherwise expected mization. However, if we approximate this optimal value (for δ sufficiently small). by taking 2k = m for even m then the cost of the Trot- The above argument only discusses the scaling of the terized simulation is O(δ−2/m), which leads to subpoly- evolution time under the assumption that m is fixed. If nomial (but not polylogarithmic) scaling of the total cost m grows as δ shrinks then there may be m–dependent of simulation in 1/δ if 2k = m and the Hamiltonian is prefactors that are ignored in the above analysis. If the sufficiently smooth because the cost of a Trotterized sim- Hamiltonian is analytic in a strip about [0, 1] and it re- ulation grows exponentially with k unlike the error in the mains gapped throughout the evolution then Lidar et al. adiabatic approximation. 16

This shows that the cost of implementing P ff 0 |0i within error δ, T0, is sub–polynomial in 1/δ. Since H Z(θ) H O(−1 log(1/)) repetitions of this circuit are needed and since errors are subadditive it follows that taking U δ ∝ 2/ log(1/) suffices to make the total error at most ff −1+o(1) . Thus the cost of implementing P0 scales as  where the o(1) term is the amalgamated costs of the adi- abatic preparation and the logarithmic term from phase FIG. 12. General phase estimation circuit to compute the estimation. We therefore can use this strategy to achieve expectation value of any unitary which can be given as a a (near) quadratic speedup without ever measuring the controlled black box. The Z(θ) produces a rotation about the Z axis by angle θ; by varying this, real and imaginary state, which is important if the simulation is used within parts of the expectation value can be measured. a larger quantum algorithm. It also should be noted that high–order Trotter–Suzuki formulas may not be needed in practice as numerical ev- and spin correlation functions idence suggests that small Trotter numbers typically suf- 1 fice for the small Hubbard models that we have consid- z z Si Sj = (ni,↑ − ni,↓)(nj,↑ − nj,↓) (26) ered. This means that low–order methods may often be 4 preferable to their higher–order brethren. Tight error can be determined directly by measuring the values of bounds for the second order Trotter–Suzuki formula are four qubits. given in the appendix. By measuring all qubits in the computational basis (eigenvectors of the Pauli-Z operator) we can measure density, double occupancy and all spin and density cor- VI. MEASURING OBSERVABLES relation simultaneously.

In this section we will discuss the measurement of in- teresting physical observables. The total energy, which 2. General strategy for other observables could be measured by phase estimation, is the least inter- esting quantity. We will instead focus on densities and Next, we note that there is a general strategy used density correlations, kinetic energies, Green’s functions to measure the expectation value of any unitary oper- and pair correlation functions. ator U, assuming that we can build a circuit that im- Such computations will allow us to understand the plements a controlled version of this unitary, controlled physics of the ground state. For instance, if the ground by some ancilla. Namely, we apply a one-bit phase es- state is superconducting, then we can compute the corre- timation using the phase estimation circuit of Fig. 12. lation function of the pair field by the methods described This is a standard trick; see for example Fig. 9 in Ref. in this section. Its long-distance behavior determines the 14. Since we have circuits that implement unitary evo- condensate fraction. We can also compute the expecta- lution under various Hamiltonian terms, this enables us tion value of the kinetic energy; its variation with doping to meaure these terms. For example, to measure a term † can help determine whether the system becomes super- cp,σcq,σ, we measure the expectation value of the unitary † † conducting as a result of kinetic energy gain compared exp[−iθ(cp,σcq,σ +h.c.)]. Since the operator cp,σcq,σ +h.c. to the non-superconducting state. Finally, more com- has eigenvalues −1, 0, +1, the most efficient results are plex correlation functions, which would be very difficult obtained from the phase estimation when we choose – if not impossible to determine in experiments – could θ = π (which perfectly distinguishes in a single mea- determine how these superconducting properties vary in surement between eigenvalue 0 and eigenvalues ±1) or response to perturbations that enhance or suppress ef- θ = π/2 (which perfectly distinguishes the case of eigen- fects such as spin or charge fluctuations. value +1 from the case of eigenvalue −1). However, for the observables we consider, there in fact are much simpler ways of measuring the correlation func- A. Local observables and equal-time correlations tions. We give two different strategies. The first strat- egy involves replacing rotations in some of our unitary 1. Measuring the density, double occupancy and spin and gates with measurements and we call it the “stabilizer density correlations strategy”; the second introduces a new gate called an “FSWAP”.

The densities ni,σ are trivially measured by measuring the value of the qubit corresponding to the spin-orbital 3. The Stabilizer Strategy (i, σ). Similarly double occupancies ni,↑ni,↓, density cor- relation functions The stabilizer strategy is a method for measuring ob- ninj = (ni,↑ + ni,↓)(nj,↑ + nj,↓) (25) servables of the form exp[−iθ(O1+O2+...)] or of the form 17

O1 +O2 +..., where each Oi is a product of some number The circuit expressed in standard gates is: of Pauli operators, and [Oi,Oj] = 0. This form includes many operators of interest to us, including terms in the Hamiltonian such as the kinetic energy as well as other terms such as pair correlation. We call this the stabilizer strategy because of our use of products of Paulis which F = commute with each other; no assumption is made that any state is a stabilizer state. H H If we can measure each Oi, then we succeed in measur- ing the desired operator. Since they commute, we may (28) measure them in any order without needing to recreate the state after measurement. As each Oi is a product of We next change the basis to be able to better mea- Paulis, there is some unitary in the Clifford group which sure the kinetic energy on the bond which is then maps it onto a Pauli Z operator on some given qubit. † †
−tpq cp,σcq,σ + cq,σcp,σ . This basis change is a simple Hence, we can measure that Oi by applying that Clifford circuit: unitary, then doing a Z-basis measurement, and finally undoing the Clifford unitary. Applying this procedure to terms in the Hamiltonian, for which we have previously given circuits to implement exp[−iθ(O1 + O2 + ...)] using sequences of controlled Z-basis rotations conjugated by H Clifford gates, the measurement circuit amounts to re- (29) placing each controlled Z-basis rotation in the evolution circuit with a Z-basis measurement. For example, if we giving the unitary: apply this procedure to the Hpq unitary evolution shown in Fig.9, we arrive at the circuit shown in Fig. 13. Sum-  √ √  1/ 2 0 0 1/ 2 ming the value of the two Z-measurements (treating the √ √ answers as ±1 for each measurement) and then divid-  −1/ 2 0 0 1/ 2   √ √  . (30) ing by two (this division occurs because the controlled  0 1/ 2 1/ 2 0  unitaries rotate by an angle equal to half the coupling  √ √  0 −1/ 2 1/ 2 0 strength) gives a measurement of the expectation oper- ator of the hopping operator. In fact, we find that the An example circuit containing several measurements same measurements are needed to measure the pairing in parallel is shown in Fig. 14. operator for ∆ real, since as we have noted, the pairing We may then measure the Z eigenvalues of the two operator for ∆ real is implemented by the same unitary qubits and call them s1 and s2. If we measure s1 = −1, as the hopping operator, up to a sign change in the sec- there is one particle on the two sites. We then have as ond controlled rotation. Thus, the two measurements we † † estimators for the Green’s function c c 0 + c 0 c : have given simultaneously measure the two commuting p,σ q,σ q,σ p,σ † † † operators cp,σcq,σ0 + h.c. and cp,σcq,σ0 + h.c. (to measure s δ (31) the pairing operator, one must instead consider the dif- 2 s1,−1 ference of the two Z measurements). † †
and for the kinetic energy −tpq cp,σcq,σ + cq,σcp,σ the estimator

4. The FSWAP Strategy for the kinetic energy and Green’s − tpqs2δs1,−1 (32) functions Note that if we have N qubits we can do N/2 measure- ments simultaneously, as long as no qubit is involved in For the kinetic energies we will have to measure one- † † two measurements. body Green’s functions ci,σcj,σ + cj,σci,σ. To do this we Conversely, if the first qubit is +1, there are either 0 first swap qubit j with its neighboring qubits until it or 2 particles on the bond. In the presence of a pair- is neighboring qubit i in the normal ordering and is at ing term, which breaks particle number conservation we one of the positions k = i ± 1. Taking into account the can measure non-trivial values for the pair field operator fermionic nature of the electrons we need to use a new † † c c 0 + h.c. as s δ . fermionic swap gate FSWAP with matrix p,σ q,σ 2 s1,1   1 0 0 0 5. The FSWAP Strategy for the pair correlation functions    0 0 1 0    . (27)  0 1 0 0  Pair correlation functions in their most general 0 0 0 −1 form are built from the general 2-body terms 18

H H Y Y †

H |Mi H Y |Mi Y †

FIG. 13. Hpq Measurement

For the Hubbard model we are interested in singlet pairing and the specific terms we want are:

1  † † † †  H ∆ij,kl = c c − c c (ck,↑cl,↓ − ck,↓cl,↑) . (36) F 2 i,↑ j,↓ i,↓ j,↑ Multiplying this out and sorting by spin we get F 1  ∆ = − c† c c† c + c† c c† c H ij,kl 2 i,↑ k,↑ j,↓ l,↓ i,↑ l,↑ j,↓ k,↓ † † † †  +cj,↑ck,↑ci,↓cl,↓ + cj,↑cl,↑ci,↓ck,↓ , (37) H which can be measured by individually measuring all F terms. Note that even though there may be N 4 different F choices for the four indices, we do not need to measure all of these four-point functions. As pairs are expected H to be tightly bound in the Hubbard model, choosing i close to j and k close to l, but choosing the pairs (i, j) and (k, l) at as large a distance as possible on a given lat- tice is sufficient to check for long-range pair correlations. FIG. 14. Example of a circuit for parallel measurements of † † † † a number of Green’s function values between different sizes In fact, we can measure ci,↑cj,↓ + h.c (or, ci,↑cj,↓ − h.c) for N/2 pairs (i, j) simultaneously because the operators commute; then, averaging the result over pairs extracts † † ci,σcj,−σck,−σ0 cl,σ. This expectation value can be mea- the long wavelength pair correlation. To measure these sured after fermion-swapping the qubits to be adjacent with minimum depth, one can use nesting strategies sim- and then performing two Green’s function measurements. ilar to those discussed for measuing hopping terms in We first use FSWAP gates to move the qubits i, k, j, Sec.IVB. and l to four adjacent positions which we will label 1 to 4. We then apply the same circuit as for the Green’s function measurements to both pairs (1, 2) and (3, 4) and B. Dynamic correlation functions and gaps measure the Z component of each qubit m (=1,2,3, or 4) as sm. The expectation value is then expressed as 1. Dynamic response in the time domain † † 0 hci,σck,σcj,σ0 cl,σ i = hs2s4δs1,−1δs3,−1i Care must be taken if two indices are the same, e.g. if The measurement of dynamic correlation functions i = k in above term. In that case things simplify since CA,B(t) = hψ0|A(t)B(0)|ψ0i (38) † † † hc c c c i = hn c c i (33) +itH † −itH i,↑ i,↑ j,↓ l,↓ i,↑ j,↓ l,↓ ≡ hψ0|e A e B|ψ0i and the measurement just becomes for time-independent Hamiltonians such as the Hubbard model are typically presented not in the time domain but † hni,↑cj,↓cl,↓i = hni,↑s4δs3,−1i (34) after a Fourier transform in the frequency domain: Z ∞ Similarly if i = k and j = l we get iωt SA,B(ω) = dtCA,B(t)e (39) † † −∞ hci,↑ci,↑cj,↓cj,↓i = hni,↑nj,↓i (35) The standard way of measuring them has been explained and the measurement is straightforward. in several references, most explicitly in Ref. 14. For 19 unitary operators A, B, the product A†e+itH Be−itH is a Instead of performing a “simple sampling” and mea- unitary, and can be measured using the circuit of Fig. 12. suring CA(t) for all times t and then Fourier-transforming One simple improvement is to note that it is not neces- it and hoping to get these delta-functions resolved a new sary to control the evolutions eitH , e−itH and it suffices and better approach is to do “importance sampling” and to control A†,B; then, if the ancilla bit is |0i so that the measure the energies of the eigenstates |ni with eigenen- † itH −itH 2 operators A ,B are not done, the product e e is ergies En directly with the weight |hψn|A|ψ0i| with equal to the identity as desired. The uncontrolled time which they appear in above sum. evolution will require only half as many arbitrary angle This can be achieved by phase estimation. If we ap- rotations as the controlled time evolution for many gate ply phase estimation to the state A|ψ0i then eigenstate 2 sets (see Sec.VA for discussion of implementing con- |ψni will be picked just with the weight A |hψn|A|ψ0i| trolled time evolution). and a histogram of the measured energies thus directly A further improvement is to note that the final evo- measures the dynamic structure factor SA(ω). lution e−itH just gives an overall phase acting on the The retarded correlation function: ground state, and so can be replaced by Z rotation of  †  the ancilla without being implemented, giving another χA(t) = θ(t)hψ0| A (t),A(0) |ψ0i (42) factor of two reduction in depth. A final improvement combines the idea in Ref. 14 of controlling one opera- can then be obtained as follows. First, we write the tion and anti-controlling another with one of the ideas retarded correlation function in terms of its real and in Sec.VA of implementing either forward or backward 0 00 imaginary parts, χA(ω) = χA(ω) + iχA(ω). The imagi- evolution in time. We prepare the ancilla in the state |+i. 00 nary part of the retarded correlation function, χA(ω) is If the ancilla is |0i, we implement eitH/2B and if the an- then obtained from the fluctuation-dissipation theorem: cilla is |1i we implement e−itH/2A. The measurement of 00 −βω χA(ω) = (1 − e )SA(ω); at zero-temperature, β = ∞, the ancilla in the X basis gives the desired expectation and the second term in parentheses vanishes. The real value. The controlled time evolution, by either eitH/2 or 0 part χA(ω) is then obtained by the Kramers-Kronig re- e−itH/2 depending on whether the ancilla is |0i or |1i, lation: can be done as described in Sec.VA. This requires half Z ∞ 0 00 the number of time steps as that required implement the 0 dω χA(ω) itH χA(ω) = P 0 (43) uncontrolled evolution by e , assuming that the same −∞ π ω − ω timestep is used for both evolutions; the only additional cost is an additional CNOT gate for each arbitrary angle By choosing A = Zp,σ we can measure the local dy- Z rotation in the evolution and since these are Clifford namical spin density correlations. Explicitly we find that gates the cost is much smaller than the cost of an arbi- trary angle rotation for many gate sets. hZp,σ(0)Zp0,σ0 (t)i = 4hnp,σ(0)np0,σ0 (t)i (44)

−2np,σ(0) − 2np0,σ0 (t) + 1.

2. Dynamic response in the frequency domain by phase Since np0,σ0 (t) = np0,σ0 (0) the last three terms do not de- estimation pend on t and only contribute a constant, which does not give any contribution to the interesting finite-ω structure factor. Local dynamical charge and spin correlations are straightforwardly calculated from the spin density corre- † CA(t) = hψ0|A (t)A(0)|ψ0i (40) lations. +itH † −itH Similarly, local single-particle excitations can be mea- ≡ hψ0|e A e A|ψ0i sured by choosing can be simplified after a Fourier transform into the fre- † Y quency domain: A = cp,σ + cp,σ = Zq,σXq,σ. (45) q

VII. NON-DESTRUCTIVE MEASUREMENTS For example, to measure a Green’s function we add a † † “hopping” term λ(cp,σcq,σ +cq,σcp,σ) to the Hamiltonian. For the small systems that we have explicitly simu- In order to obtain an estimate with an error  we need lated, the time needed to adiabatically prepare a good to choose λ = O(), and then measure the energy to an 2 estimate of the ground state is small compared to the accuracy λ = O( ). Phase estimation then requires −2 phase estimation time. However, this will change as one time O( ), which scales the same as repeated prepara- increases the system size, and the time to adiabatically tion with destructive measurements and thus only gets a prepare the ground state will ultimately dominate, at constant improvement. least if one uses an annealing path that linearly interpo- The scaling can be improved by using a symmetric lates between initial and final Hamiltonian. The reason estimator for the derivative is the time to adiabatically prepare the ground state is −2 E0(λ) − E0(−λ) 3 expected to scale as ∆ for a system with spectral gap hΨ0(0)|O|Ψ0(0)i = + O(λ ), (48) ∆, while resolving energies to accuracy ∆ with phase es- 2λ timation takes time 1/∆ and thus for small ∆ the phase which reduces the approximation error. To get the en- estimation is faster. We note that the improved higher- ergy and thus allows a larger value of λ = O(1/2), order annealing paths inVC do alleviate this problem. which requires phase estimation only to an accuracy of This problem is slightly alleviated by the fact that we O(3/2), and a time scaling as O(−3/2). In general, are content to prepare by annealing a state with signif- using a k-th order estimator for the derivative requires icant overlap (say, 1/2 overlap) with the ground state, O(k) energy measurements, but with an error scaling as as then phase estimation will give us a significant chance O(λk), we can choose λ = O(1/(k−1)), and require time of projecting onto the ground state. However, once we O(k−1−1/(k−1)). Asymptotically for small  we can thus have prepared a ground state ψ0 by a combination of an- estimate the expectation value non-destructively with an nealing and phase estimation, it will be desirable to mea- log  effort scaling as O(−  ), obtaining a near-quadratic sure properties of the state without destroying it. Here speedup. we present two approaches for such non-destructive mea- In the Hellman-Feynman approach, there may be no surements. SectionVIIA discusses an approach based on need to do an adiabatic evolution. In many cases we the Hellman-Feynman theorem, which not only is non- could simply start in the state ψ0, then apply a phase destructive but also scales better in the error than the estimation to expectation value of H + λO, and then simple way of repeatedly preparing the ground state and apply another phase estimation to estimate the ground measuring . An alternate approach based on “recover- state energy of H. The absolute value squared of the ing” the state after measurements may be more useful in overlap between the ground state of H and the ground some circumstances and is discussed in Sec.VIIB. state of H + λO is equal to 1 − O(λ2/∆2), and so for λ << ∆ the overlap is close to unity. So, with probability close to 1, the energy given by phase estimation of H+λO A. Hellman-Feynman based approach will in fact be the desired ground state energy, and with again probability close to 1 the second phase estimation According to the Hellman Feynman theorem the will return to the ground state of H. derivative of the ground state energy E0(λ) with respect to a perturbation λO is just the expectation value of the operator O in the ground state |Ψ0(λ)i: B. Non-Destructive Measurements With Quadratic Speedup d d E0(λ) = hΨ0(λ)|H + λO|Ψ0(λ)i dλ dλ This annealing procedure becomes less useful for op- = hΨ0(λ)|O|Ψ0(λ)i. (46) erators O which require a complicated quantum circuit. An alternative and superior way of measuring expecta- Suppose we wish to measure the expectation value of z z tion values of arbitrary observables O is to adiabatically a time dependent correlation such as O = S (t)S (0), z z add a small perturbation λO to the Hamiltonian H. where t > 0 and where S (t) = exp(iHt)S exp(−iHt). This opens a way to measuring observables without In this case, the quantum circuit to measure O (given below) is much more complicated; thus the Hamiltonian destroying the ground state wave function |Ψ0(0)i of H. We adiabatically evolve the ground state wave function H + O becomes significantly more costly to do phase estimation on. |Ψ0(0)i to |Ψ0(λ)i by slowly increasing λ to its final (small) value, and then perform another quantum phase In this section, we explain a technique to measure ar- bitrary projection operators in a non-destructive fash- estimation to determine the energy E0(λ). The ground state expectation value can be estimated through ion. We then further explain a quadratic speedup of this approach. Our initial approach will give many in- d E (λ) − E (0) dependent binary measurements with outcome probabil- hΨ (0)|O|Ψ (0)i = E (λ)| = 0 0 +O(λ2). 0 0 dλ 0 λ=0 λ ity determined by the expection value of the operator. (47) This will require a number of samples proportional to 21 the square of the inverse error. On the other hand, the Then, the probability that the subsequent measurement improved approach will exploit phase estimation to speed of Q gives outcome j is equal to this up quadratically (up to log factors). 2 As we have explained previously, we have quantum cir- |δi,j − ajai| 2 , (52) cuits that implement projective measurements of all the 1 − |ai| operators we are interested in, such as number operator, and the resulting state is exactly equal to φj. Repeating hopping operator, and so on. Further, we have a general this, we find that after every measurement of Q with technique for implementing a projective measurement of outcome i, the probability that the measurement of P0 any unitary that can be controlled. 2 will return to the ground state is |ai| , while if we do not return to the ground state the probability that the next measurement of Q will give outcome j is given by 1. Recovery Map Eq. (51). The transitions are then governed by a Markov chain Suppose we wish to implement a projective measure- with transition probabilities ment with k possible outcomes, for some given k. We 2 2 2 describe these outcomes by projectors Q1, ..., Qk, with Tj←i = 1 − 2|ai| + |ai| |aj| , (53) P i Qi = I, and we refer to this measurement as “mea- suring Q”. The ground state ψ0 can be written as a T = |a |2, (54) linear combination of states in the range of each of these 0←i i projectors: where Tj←i is that, starting in state φi, a measurement X of P0 reveals that one is not in the ground state, and ψ = a φ , (49) 0 i i then a measurement of Q gives outcome j, while T0←i is i the probability that, starting in state φi, does leave on where in the ground state. The initial measurement of Q at the start of the Markov chain gives state φi with probability 2 Qiφi = φi, (50) |ai| . We would like to work out the expected number of measurements before the algorithm terminates back in and |φi| = 1. Assume for the moment that we can also the ground state. implement a measurement P0 which projects onto the To do this calculation, we re-interpret the probabili- ground state ψ0 (we describe how to do this to sufficient 2 ties, saying that with probability 2|ai| the system makes accuracy below). Then, the algorithm for nondestruc- some transition; half the time this transition leads to tive measurement is: first, begin in the ground state. state 0, while the other half the time the system makes Then, measure Q. Then, measure P0. If the result is a transition from state i to some state j, and state j is that one indeed is in the ground state, then we have 2 chosen with probability proportional to state |aj| (note restored the ground state; at this point we can either that j may equal i, so some of the “transitions” do not re-measure the projector Q to obtain better statistics, actually lead to a change in state). This re-interpretation or make some other measurement of a different observ- 2 only makes sense if |ai| ≤ 1/2, of course but it will al- able. If instead the measurement of P0 reveals that we low us to compute the expected time exactly and as the are not in the ground state, we re-measure Q and then expected time is analytic, this calculation will then work re-measure P0. We repeat this process of re-measuring 2 for all |ai| . The advantage of this re-interpretation is Q and re-measuring P0 until we are in the ground state. that if a transition occurs, and if the transition leads to The basic idea is similar to that in Ref. 44. a state other than 0, then the probability that that state The rest of this subsubsection is focused on calculat- 2 is equal to j is proportional to |aj| , which is the same ing the probability of returning to the ground state; most probability distribution as the chain began in. readers may wish to skip to the next subsubsection, which Starting in state i, the expected number of steps before gives the more important quadratic speedup. The main 2 a transition is 1/2|ai| . Thus, averaging over states i with reason for introducing the recovery map in this subsub- 2 initial probability distribution |ai| , the expected number section is that it can be used after the quadratic speedup of steps before a transition is k/2, where k is the number (see below). Let us compute the probability of returning of possible outcomes of measurement Q. Half of these to the ground state in this process. If the initial mea- transitions return to state 0, while the other half do not, surement of Q gives outcome i then the resulting state 2 2 2 leading to some state j 6= 0 with probability distribution is φi; since, |hψ0|φii| = |ai| , we have a probability |ai| 2 proportional to |aj| . Thus, the average number of steps of returning to the ground state after measuring P0. We before returning to the ground state is now analyze the case that this measurement of P0 reveals that we are not in the ground state; then the resulting Nsteps = k. (55) state is equal to Note that both the number of phase estimations required 1 and the number of measurements of Q required are equal (1 − P0)φi. (51) p 2 1 − |ai| to k. 22

2. Quadratic Speedup revealing additional information. If instead the measure- ment gives several bits of information on the energy of the Suppose we wish to measure a projector Q. Define state, the procedure does not work; intuitively, each φi is a coherent superposition of different energy eigenstates unitaries U = 2Q − 1 and V = 2P0 − 1. Assume that P0 and making this more detailed measurement destroys this has rank one. Applying any sequence of projections P0 information. or Q to the ground state ψ0 gives some state in the space spanned by the non-orthogonal basis vectors ψ0, Qψ0. We now describe how to do this; we call this proce- Hence, we will work in this two-dimensional space for dure a “coherent phase estimation”. We have the ability our analysis (the degenerate cases that Qψ0 = ψ0 or to perform a phase estimation on the Hamiltonian H to Qψ0 = 0 can be handled easily too and one can verify determine the energy of a state. This phase estimation is that the final result will be correct in these cases also). In given by a quantum circuit, in which several control bits this subspace, in the non-degenerate case, we can write are initialized in a state |0i. Then, a Hadamard is ap- plied to each control bit. Then, the control bits are used ! 1 0 to control the application of the Hamiltonian for a set V = , (56) period of time. Finally, the Hadamard is re-applied, and 0 −1 the control bits are measured in the computational basis. Then, classical post-processing is applied to extract the ! energy of the state. This process can be performed in se- 2 cos2(θ) − 1 −2 cos(θ) sin(θ) U = , (57) rial (using only one control bit) or in parallel; the parallel −2 cos(θ) sin(θ) 2 sin2(θ) − 1 approach uses more control bits but reduces the depth of the circuit. The measurement of the energy is not de- 2 where the angle θ is such that hψ0|Q|ψ0i = cos (θ). We terministic; however, one can employ a larger number of emphasize that the above two equations are written in control bits to increase the accuracy. 2 the orthogonal basis ψ0,Z(Qψ0 − cos (θ)ψ0), where Z To measure P0, we use this phase estimation in its is some normalization factor. Thus, the unitary UV re- parallel form, but do not measure the outcomes of the stricted to the subspace has eigenvalues control bits. We initialize an additional “outcome” bit to |0i. We then use a unitary quantum circuit to im- exp(±2iθ). (58) plement the classical postprocessing done to determine the energy, storing that energy value coherently. Then We now build a quantum circuit that adds an addi- a unitary quantum circuit determines if that energy is tional ancilla that controls both U and V , hence controls near the ground state value (which we have determined whether or not UV is applied. We then implement phase in advance by a phase estimation before doing any mea- estimation using this ancilla to determine the eigenvalues surements), and if so, it flips the value of the outcome of UV . We use the ground state ψ0 as input to the phase bit. Finally, we uncompute, reversing the classical post- estimation circuit. Let T0 be the time required to imple- processing and phase estimation steps, and then at the ment UV . Using the ancilla to control (UV )n, for various end we measure the outcome bit. This procedure is es- powers of n, one is able to measure the eigenvalue of UV sentially that in Ref. 44. While this procedure does not −1 −1 to accuracy  in a time proportional to T0 log( ); in implement a projective measurement, but rather imple- this case, we take n of order −1. ments a POVM, if sufficient numbers of control bits are In fact, UV has two distinct eigenvalues, and the phase used, the measurement can be arbitrarily close to the estimation will return one of the two answers randomly. desired projective measurement.

However, since the eigenvalues differ only in sign, and the Performing this measurement of P0 does require addi- 2 desired expectation value hψ0|Q|ψ0i is equal to cos (θ), tional qubits. The number of qubits required scales pro- both eigenvalues give the same answer for this expecta- portional to the number of bits of accuracy desired; how- tion value. ever, since this accuracy is of order ∆, if ∆ is only poly- After implementing this phase estimation to the de- nomially small in system size then this requires only log- sired accuracy, we then are left with some state in the arithmically many extra bits. Additionally, the more ac- given subspace. If we then wish to recover the ground curately we implement a projective measurement rather state (to recover some other observable), we can apply than a POVM, the more control bits required, but this the recovery map above by alternating measurements of overhead also only scales logarithmically. Q and P0 until the ground state is restored. An alternate method of implementing P0 is to note that we can easily implement the projector onto the ground state of the Hamiltonian at the start of the annealing pro- 3. Measuring P0 cess if our initial Hamiltonian is of a sufficiently simple form, such as a free fermion Hamiltonian or a Hamil- The above procedure relies on the ability to measure tonian with a product ground state. For a general free P0. The key is to define an operation that will only iden- fermion Hamiltonian, we can use Givens rotations to ro- tify whether or not one is in the ground state, without tate to the case that the ground state simply has some 23 spin-orbitals occupied and some empty; we then mea- ground states of various candidate mean-field Hamiltoni- sure whether we have the desired pattern or not. Call ans, from which an adiabatic evolution to the ground ff this projector P0 . Then, if Uadiab is the unitary which state of the Hubbard model can be attempted. describes the approximately adiabatic evolution from the To implement time evolution under the Hubbard and free Fermion Hamiltonian onto the desired final Hamilto- mean-field Hamiltonians we have given explicit quantum ff † circuits for all terms, and discussed the size of the Trot- nian, we can approximate P0 ≈ UadiabP0 Uadiab. The ac- curacy of this approximation depends upon the annealing ter time step required to achieve sufficiently small er- path; see the discussion inVC. The improved annealing rors. We discuss how time evolution under the individ- paths discussed there are most useful for the particular ual terms in the Hubbard model can be efficiently paral- approach here where we need very high accuracy in the lelized, ultimately requiring O(N) qubits and gates with annealing; in all other applications in the paper, we need only O(log N) parallel circuit depth for one time step, only moderate accuracy in the annealing, sufficient to which allows efficient simulations of very large systems. get a large overlap with the ground state so that phase We gain additional constant factors over previous ap- estimation can then project onto the ground state with proaches, by optimizing the phase estimation algorithm reasonably large probability. to reduce the required number of (expensive) rotation gates by a factor four. We furthermore propose to use a larger Trotter time step for the adiabatic state prepa- ration (whose time scale is controlled by the inverse gap VIII. CONCLUSIONS squared), to prepare a very good (but not perfect) guess for the ground state, and then refine this to the exact In this paper we have presented a comprehensive strat- ground state by doing only the final quantum phase esti- egy for using quantum computers to solve models of mation (whose time scale is shorter since it is controlled strongly correlated electrons, using the Hubbard model by the inverse gap) with a small time step. as a prototypical example. Similar strategies can be used We have finally discussed approaches to obtain a for generalizations of the Hubbard model and other dis- quadratic speedup in measurements by proposing two crete quantum lattice models, including, but not limited non-destructive measurement strategies, one based on to, the t-J model, frustrated magnets, or bosonic mod- the Hellman-Feynman theorem, and another based on els coupled to (static) gauge fields, which all suffer from recovering the ground state after a (destructive) mea- negative sign problems in quantum Monte Carlo simu- surement of a single qubit. We also introduce a new lations on classical computers. We go beyond previous approach of measuring dynamic structure factors and papers6,12–14 that discussed simulations of the Hubbard spectral functions directly in frequency space, using ideas model on quantum computers by giving complete details similar to angle-resolved photoemission experiments. of all steps needed to learn about the properties of the Estimating the gate counts required to simulate the Hubbard model. Hubbard model, we find that even on lattices with more In particular we presented a strategy to prepare trial than N ≈ 1000 sites – which should be large enough to ground states starting from various mean-field states learn most interesting properties – can be simulated on or RVB states on decoupled plaquettes. While finding small scale quantum computes using only a few thousand the ground state of quantum lattice models is QMA- qubits and a parallel circuit depth of about one million hard3,45,46 this is a statement about the worst case. We gates. This number is based on our estimate of a few expect that, as experience has shown, many experimen- thousand time steps with a circuit depth of not more tally relevant models have ground states with either no than a few hundred gates each. This means that even long-range order (so-called quantum liquids) or with bro- with logical gate times of the order of 1µs, the ground ken symmetries that are qualitatively well described by state of the Hubbard model can be prepared within a few mean-field theories. In fact, if a model has such pecu- seconds. Small scale quantum computers will thus be a liar properties that it is hard to find its ground state very powerful tool for the investigation of many problems on a quantum computer, then also a material described in the field of strongly correlated electron models that are by that model will have a hard time reaching its ground currently out of reach of any classical algorithm. state. Quick low-temperature thermalization in experi- ments is thus an indication that the ground state of the model describing its properties should also be easy to pre- ACKNOWLEDGMENTS pare on a quantum computer. The opposite is the case, e.g. for (classical) spin glasses, where finding the ground We acknowledge useful discussions with Bela Bauer. state is NP-hard but also spin glass materials never ther- This work was supported by Microsoft Research, by the malize nor reach the ground state. Swiss National Science Foundation through the National We have furthermore presented an efficient and deter- Competence Center in Research NCCR QSIT, and the ministic quantum algorithm to prepare arbitrary Slater ERC Advanced Grant SIMCOFE. The authors acknowl- determinants as initial state, which scales better than the edge hospitality of the Aspen Center for Physics, sup- algorithms of Refs.6 and 14. This can be used to prepare ported by NSF grant # 1066293. 24

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In the above equation, we have corrected this derivatives of the Hamiltonian are small relative to an error, so that the right hand-side is a sum of two distinct appropriate power of the minimum eigenvalue gap. This norms for each j, rather than a norm of a sum. means that in some circumstances adiabatic evolution To show Eq. (A3), we write H0,H00, ... to denote the may actually involve a rapid passage. Furthermore, since first, second, ... derivatives of H with respect to time. By we are interested in high accuracy state preparation even Taylor’s theorem, H(u) = H(1/2) + (u − 1/2)H0(1/2) + R u 00 −i R 1 H(u)du though such errors may be small they will not necessarily (u − s)H (s)ds. Hence T e 0 is be negligible compared to our target error tolerance. 1/2   In this section we analyze these errors in more detail. −i R 1 H(1/2)+(u−1/2)H0(1/2)+R u H00(s)(u−s)ds du In particular, we will examine the error in the second T e 0 1/2 , order order Trotter–Suzuki formula and thus m 1 R 1   −i H(u)du Y −iH (1/2)/2 Y −iH (1/2)/2 R 1 0 0 j j 1 −i H(1/2)+(u−1/2)H (1/2) du T e ≈ e e , (A1) −i R H(u)du 0 T e 0 − T e j=1 j=m where we approximate the evolution from time 0 to 1 by Z 1 Z u 00 a single second-order Trotter-Suzuki step. The bounds ≤ H (s)(u − s)ds du 0 1/2 that we present are a generalization of those in Ref.9 to 1 00 the time–dependent case. Bounds for the Trotter formula ≤ maxskH (s)k. (A5) are given in Ref 47. 24 Our main result is that We next bound the difference m 1   R 1 0 −i R 1 H(u)du Y −iH (1/2)/2 Y −iH (1/2)/2 −i H(1/2)+(u−1/2)H (1/2) du T e 0 − e j e j T e 0 − e−iH(1/2) .

j=1 j=m 1 00 1 0 Transforming to the interaction representation gives ≤ maxskH (s)k + k[H (1/2),H(1/2)]k 24 12      −i R 1 H(1/2)+(u−1/2)H0(1/2) du m 0 X X T e + Hj(1/2), Hk(1/2) ,Hj(1/2) 1    −i(1/2)H(1/2) −i R (u−1/2)H0 (1/2)du 0 u j=1 k>j = e T e    × e−i(1/2)H(1/2), X X +  Hk(1/2),Hj(1/2) , H`(1/2) . (A2) where we define k>j `>j 0 −i(1/2−u)H(1/2) 0 i(1/2−u)H(1/2) We prove the bound in two steps. First, we will show Hu(1/2) ≡ e H (1/2)e . that So by a unitary rotation −i R 1 H(u)du −iH(1/2) kT e 0 − e k (A3)   −i R 1 H(1/2)+(u−1/2)H0(1/2) du 0 −iH(1/2) 1 00 1 0 kT e − e k (A6) ≤ maxskH (s)k + k[H (1/2),H(1/2)]k. 24 12 −i R 1(u−1/2)H0 (1/2)du = kT e 0 u − 1k Then, we use the bound of Ref.9 that R 1 0 R 1 0 −i 0 (u−1/2)Hu(1/2)du −i 0 (u−1/2)H (1/2)du m 1 = kT e − T e k. Y Y e−iH(1/2) − e−iHj (1/2)/2 e−iHj (1/2)/2 0 0 Note that kHu(1/2) − H (1/2)k can be written as j=1 j=m Z 1      −i( 1 −u)(1−x)H( 1 ) 0 1 i(1/2−u)(1−x)H( 1 ) m ∂x e 2 2 H ( )e 2 dx X X 0 2 ≤ Hj(1/2), Hk(1/2) ,Hj(1/2) (A4) 0 0 j=1 k>j which can be used to show that kHu(1/2) − H (1/2)k ≤    |u − 1/2|k[H0(1/2),H(1/2)]k, and so X X −i R 1(u−1/2)H0 (1/2)du −i R 1(u−1/2)H0(1/2)du +  Hk(1/2),Hj(1/2) , H`(1/2) . 0 u 0 kT e − T e k k>j `>j Z 1 2 0 Eq. (A2) then follows from the triangle inequality. We ≤ |u − 1/2| k[H (1/2),H(1/2)]kdu remark that in Ref.9, a typographical error led to 0 1 k[[A, H(x)],H(x)]k ≤ k[[A, B],A]k + k[[A, B],B]k be- = k[H0(1/2),H(1/2)]k. (A7) ing replaced by k[[A, H(x)],H(x)]k ≤ k[[A, B],A]] + 12 [[A, B],B]k in the language of that paper in appendix Eq. (A3) then follows from Eqs. (A5,A6,A7) and the B (see text above Eq. (B3) of that paper). This pair of triangle inequality.