Ultrafast State Preparation via the Quantum Approximate Optimization Algorithm with Long Range Interactions

Ultrafast State Preparation via the Quantum Approximate Optimization Algorithm with Long Range Interactions

Wen Wei Ho Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA    Cheryne Jonay Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada    Timothy H. Hsieh Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
July 31, 2019
Abstract

State preparation protocols ideally require as minimal a circuit depth as possible, in order to be implemented in near-term quantum devices. Motivated by long range interactions (LRI) intrinsic to many experimental platforms (trapped ions, Rydberg atom arrays, etc.), we investigate the performance of the quantum approximate optimization algorithm (QAOA) with LRIs for the preparation of non-trivial quantum states. We show that this approach leads to extremely efficient state preparation: for example, Greene-Horne-Zeilinger (GHZ) states can be prepared with circuit depth, and a quantum critical point of the long range transverse field Ising model (TFIM) can be prepared with fidelity on a 100 qubit system with only one iteration of QAOA. Furthermore, we show that QAOA with LRIs is a promising route for exploring generic points in the phase diagram of the long-range TFIM. Our approach thus provides concrete, ultrafast protocols for quantum simulators equipped with long range interactions.

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Rapid experimental progress in the precise control of synthetic quantum systems, like trapped ions Blatt and Roos (2012); Zhang et al. (2017a); Islam et al. (2013), ultracold atoms Greiner et al. (2002); Bloch et al. (2008); Bernien et al. (2017) and superconducting qubits Devoret and Schoelkopf (2013); Gambetta et al. (2017), has unlocked the potential for quantum computation and simulation Schreiber et al. (2015); Gambetta et al. (2017), quantum sensing and metrology Leibfried et al. (2004); Giovannetti et al. (2004); Degen et al. (2017); Choi et al. (2018a), and also the simulation of quantum many-body phases of matter Greiner et al. (2002); Aidelsburger et al. (2013, 2014); Islam et al. (2015); Choi et al. (2016); Smith et al. (2016); Choi et al. (2017); Zhang et al. (2017b); Choi et al. (2018b). These tasks require the ability to create, with good fidelity, quantum states containing nontrivial entanglement such as Greene-Horne-Zeilinger (GHZ) states and topologically ordered states. A central challenge is therefore finding efficient state preparation protocols: ideally, these protocols should have as minimal a circuit depth as possible to be realistically implemented in such quantum simulators, in order to suppress the errors that accumulate during runtime.

Recently, the quantum approximate optimization algorithm (QAOA) has emerged as one such promising candidate for a quantum state preparation protocol Farhi et al. (2014); Farhi and Harrow (2016). In short, the QAOA is a hybrid quantum-classical bang-bang protocol that specifically incorporates feedback: it takes as an input a set of angles that specifies the durations for which time evolution between two different Hamiltonians is alternated between, in the quantum simulator. Then, measurements are performed and the subsequent optimization of a cost function (e.g. energy, by means of a classical computer) yields a new set of angles which is fed back into the simulator. Such QAOA-type circuits, with spatially local Hamiltonians, have been shown to be able to transform trivial product states into GHZ, quantum critical, and topologically ordered states, with perfect fidelity and circuit depths scaling as where is the system’s linear dimension Ho and Hsieh (2018). Conceptually, the QAOA is an example of a “shortcut to adiabaticity”, a direction in quantum state control that is actively being researched Demirplak and Rice (2003, 2005); Berry (2009); Agarwal et al. (2017); Sels and Polkovnikov (2017), as its operating principle is fundamentally different from conventional adiabatic preparation schemes Farhi et al. (2000, 2001); Pontryagin (1987); Stengel (1994); Brif et al. (2014); Yang et al. (2017).

While a circuit depth scaling as is efficient from a theoretical standpoint – there exist fundamental speed limitations imposed by Lieb-Robinson bounds Lieb and Robinson (1972); Nielsen (2006); Bravyi et al. (2006); Hastings (2010) constraining QAOA-circuits utilizing spatially local Hamiltonians, it still presents challenges experimentally, especially in terms of scalability to a large number of qubits. This motivates the search for alternative ultrafast, or extremely low-depth, circuits. A natural way to overcome these speed limitations is to utilize long-range interactions (LRI): in principle, even finite-depth circuits can prepare long-range correlated states, as entanglement and correlations can be built up between distant parts of the system in finite time Matsuta et al. (2017); Foss-Feig et al. (2015); Tran et al. (2018). Moreover, LRIs are naturally present in various experimental quantum simulator platforms, e.g. trapped ion systems (Coulomb interactions), Rydberg atom arrays (van der Waals interactions), etc., and are hence a readily accessible resource Richerme et al. (2014); Zhang et al. (2017a); Bernien et al. (2017).

To this end, in this work we explore how efficiently the QAOA protocol with long range interactions can prepare non-trivial quantum states. Specifically, we consider quantum simulators that realize long-range Ising interactions with tuneable range , motivated by trapped ion experimental setups. We find that the QAOA protocol with LRIs can prepare GHZ and quantum critical states with depth. In particular, in the limit of extremely long-range interactions , the GHZ state can be prepared with only one (two) iteration of QAOA for odd (even) system sizes. Furthermore, the quantum critical point of the Lipkin-Meshkov-Glick model Lipkin et al. (1965) can be prepared with very high fidelity and low circuit depth (e.g. fidelity for 100 spins after one iteration). We also analyze how efficiently QAOA with long but finite range interactions can prepare points within the phase diagram of the long-range transverse field Ising model. Our results thus demonstrate the utility of extremely low depth QAOA-protocols with LRIs for near-term quantum simulators to realize nontrivial many-body states of interest.

QAOA-type protocol. — We quickly recapitulate the QAOA Farhi et al. (2014); Farhi and Harrow (2016), which we modify for our purposes. Our aim is to prepare a target state with as high fidelity as possible. Usually, will be taken to be the ground state of a certain Hamiltonian, called the target Hamiltonian .

The QAOA starts off with an easily prepared initial state such as an unentangled product state, for example the ground state of a simple, paramagnetic Hamiltonian , which we will take from here on out. One then time evolves in an alternating fashion between and an “interaction Hamiltonian” denoted for a total of pairs of times:

(1)

with evolution times given by the angles respectively. We label this protocol as QAOA; it has a circuit depth .

As the goal is to closely approximate the ground state of , one can seek to find the evolution times which minimize a given cost function , usually taken to be the energy with respect to :

(2)

As can be easily seen, increasing the circuit depth of the QAOA circuit can only improve the minimal value , i.e. .

In practice, such a protocol can be implemented in a hybrid setup involving a quantum simulator and a classical computer: one first feeds the quantum simulator an initial seed of angles, which produces a certain state . Then, leveraging upon single-site accesibility in near-term quantum platforms, one measures correlations within the state and determines the cost function (2), e.g. the global energy. A classical computer is then used to obtain the next set of angles to be fed into the quantum simulator, by means of an optimization algorithm such as gradient descent or a similar protocol. The entire process is then repeated until either the global minimum is found, or a desired energy/fidelity threshold is attained.

The choice of interaction Hamiltonian is motivated by the structure of the target Hamiltonian . For example, consider targeting the ground state of the one-dimensional transverse field Ising model (TFIM) with nearest neighbor interactions:

(3)

where parameterizes the field strength and is the number of qubits. Given this target Hamiltonian, a natural choice for the interaction Hamiltonian is thus . In this case, the QAOA protocol is guaranteed to work in the limit of for any finite size system (there always exists a finite gap), as an asymptotically slow adiabatic preparation scheme can always be trotterized to the form (1). However, non-trivial behavior and an improvement over adiabaticity can arise for small , the regime of validity for experimental systems. Indeed, in a previous work, it was shown that in this specific example, the resulting protocol at depth (QAOA), can target with perfect fidelity the ground state of the model at (GHZ and quantum critical state, respectively) Ho and Hsieh (2018). It was further conjectured and supported with numerical evidence that this result generalizes to all points in the phase diagram of the model.

QAOA with LRIs. — Despite impressive progress in improving the fidelity of single qubit and two qubit gates, near-term synthetic quantum systems nevertheless have a lack of fault tolerance, and so they are prone to increasing errors with increasing circuit depth. Thus, even a circuit depth scaling as poses challenges to scalability to large number of qubits in such platforms. Fundamentally, such scaling is dictated by speed limits due to information propagation bounds (specifically, Lieb-Robinson bounds) in systems with local interactions – in short, in order for the pattern of correlations in a quantum state to change dramatically, spatially distant parts of the system must get entangled; however, local interactions give rise to a linear light cone of information propagation, and so the time taken in a finite system of size must be at least .

Long-range interactions (LRIs) have less stringent speed limits Tran et al. (2018) and can thus dramatically speed up state preparation protocols. We now show that QAOA with LRIs is a viable method for efficiently targeting nontrivial quantum states.

Specifically, consider quantum simulators where long-range Ising interactions (with power-law interactions ) can be realized, such as trapped ions experiments. Together with a readily applicable transverse field, effective Hamiltonians of the form

(4)

can be created in these setups, where and can vary in principle between 0 and 3. We have chosen the normalization of (4) such that and . Note that with this choice, the regime reduces to the nearest neighbor TFIM model with open boundary conditions.

In the following, we will analyze the small regime of the Hamiltonian (4) and show that low, depth QAOA circuits are sufficient to prepare its ground states. Given the structure of the Hamiltonian, it is thus natural to choose for the QAOA protocol

(5)

We will restrict the QAOA parameter space to and . The former is motivated by experimental limitations on the evolution time, and the latter is because which is conserved throughout the evolution.

Figure 1: Fidelity in preparation of LMG critical state as a function of system size , for QAOA depth . Remarkably, even at , the fidelity is very close to unity .

GHZ & quantum critical state preparation in the Lipkin-Meshkov-Glick model, . — We first consider the case of (4), in which the qubits interact in an all-to-all fashion. Then, up to an overall multiplicative factor and also an inconsequential shift in energy, (4) can be seen to be equivalent to the Lipkin-Meshkov-Glick (LMG) model

(6)

where the total spin operators are and , and . As is the case with the nearest-neighbor TFIM model, the ground states are ferromagnetic GHZ states at , and a quantum phase transition at separates the ferromagnet from the paramagnetic phase.

We now show that a QAOA circuit suffices to produce the ground state of (6) at , i.e. the GHZ state, for odd system size . To see this, we explicitly derive the energy cost function (2) for the LMG model with QAOA, which yields:

(7)

(see SOM () for the derivation). From the above, it is evident that for odd , the ground state energy of , namely , can be achieved with the angles . Thus, in such cases, we see that the ferromagnetic GHZ state, a state with macroscopic superposition of entanglement , can be created with two operations:

(8)

We note that there exist various preparation schemes that create macroscopic GHZ states, one of which is the Molmer-Sorenson (MS) protocol involving time evolution with Sørensen and Mølmer (2000); Monz et al. (2011). Although the QAOA protocol discussed above resembles the MS protocol, there are several differences: MS begins with the (Ising symmetry broken) ground state and prepares a GHZ state whose relative phase between the cat states depends on system size. In contrast, since the Ising symmetry is always conserved during the implementation of the QAOA protocol and since we begin with the symmetric state , the end result is that we will always prepare the symmetric GHZ state, independent of system size.

The distinction between MS and our protocol is most manifest for even system sizes, in which we find that the GHZ state is instead achieved with perfect fidelity with a QAOA protocol:

Note that from (7), there is no range of parameters that give perfect fidelity for for even . We show in the appendix SOM () the derivation of the above result. These results already demonstrate the utility of LRIs with QAOA: they enable an ultrashort, depth circuit to prepare with perfect fidelity a macroscopic GHZ state.

Besides the GHZ state, we find that the QAOA approach with LRIs is general and can target many other interesting states. One state of particular interest is the quantum critical point of the LMG model at and is a highly correlated state . By numerically optimizing for the energy of (6) at , we find that the QAOA protocol with only is already sufficient to achieve the critical state with extremely high fidelity , even for very large system sizes ( at 100 qubits) (see Fig. 1). This is a remarkably efficient protocol for preparing a quantum critical state.

Note that the gap at the critical point of the LMG model scales as Lipkin et al. (1965), and thus the adiabatic algorithm requires time to prepare the GHZ and critical states. However, in order to make a direct comparison with the adiabatic algorithm, we would need to scale down the exchange interaction in Eq. 5 by , and thus the total preparation time in this convention would scale with . Our intention is not to make this theoretical comparison, but instead to make contact with existing experiments. In recent experiments, the nearest-neighbor exchange interaction does not necessarily decrease with system size; for example, it is for respectively Zhang et al. (2017a). Hence, the interactions in Eq. 5 are reasonable in near-term trapped ion experiments and lead to preparation time for the GHZ and critical states. The simplicity and discreteness of the protocols we have presented may offer advantages over the adiabatic algorithm with or without counter-diabatic terms del Campo (2013); Campbell et al. (2015); Sels and Polkovnikov (2017); Campbell and Deffner (2017).

Figure 2: (Left) Fidelity in preparation of GHZ state as a function of system size , for . One sees that with increasing , the fidelity decreases. However, this can be compensated by going to higher s. (Right) Fidelity in preparation of GHZ state as a function of , for . One sees that generically, interactions that are longer-range (smaller ) are more helpful in targeting the desired state.

GHZ preparation with finite . – In practice, there may be challenges in realizing strictly all-to-all () Ising interactions, and therefore we analyze how well the finite QAOA protocol can prepare the GHZ state.

In the left panel of Fig. 2, we fix and show the fidelity with GHZ state achieved for QAOA for even system sizes. As expected, QAOA no longer prepares the state with perfect fidelity unlike the case (for any system size), but this can be addressed with further iterations of QAOA (note the high fidelities achieved by for system sizes up to ). The long-range interactions establish correlations between spins separated by a distance in time , which surpasses the light cone bound for local interactions Matsuta et al. (2017); Foss-Feig et al. (2015); Tran et al. (2018). Hence, we expect that the depth required to prepare the state with some fixed error in fidelity scales as . Also plotted is the optimal fidelity for different values of and for fixed system size . These results indicate that longer interaction range (smaller ) yield higher fidelities, and errors can be effectively reduced using further QAOA iterations 111We note that the numerical optimization in our simulations may output local minima in the cost function, and thus the results presented in Fig. 2 are lower bounds on the optimal fidelities..

Exploring the phase diagram of the long-range TFIM. — Finally, we explore how well the QAOA protocol can prepare the ground states at generic points in the phase diagram of the long-range TFIM model (4). Plotted in Fig. 3 are the fidelities as a function of transverse field and interaction range , for and for .

It is known that the long-range TFIM (3) supports a ferromagnetic-paramagnetic ground state quantum phase transition for each value of , upon tuning . In the limits , the critical magnetic field , while in the limit , the critical magnetic field . For intermediate values of , previous works have attempted to map out how varies (see e.g. Jaschke et al. (2017); Fey and Schmidt (2016) and Koffel et al. (2012); Vodola et al. (2016) for the antiferromagnetic model). As expected, from Fig. 3, at the system sizes considered, the QAOA with LRIs is able to target ground states within the paramagnetic phase (large ) relatively easily, while for the ferromagnetic phase (small ) it becomes more difficult to prepare as the interactions become more short-ranged (larger ). Note that for larger , the region where state preparation is difficult (red region) is separated from the region where state preparation is easy (blue region) near the value , which agrees with the critical points of the nearest-neighbor TFIM, which is realized in the asymptotic limit .

Figure 3: Heat map depicting fidelities in the state preparation of the ground state of the long-range TFIM (3) for system sizes , and for 12, . Blue (red) regions demarcate regions of high (low) fidelity. As noted before, there is an odd-even system size distinction near , for , but this distinction vanishes at the next level, .

As the small regime of this model is somewhat challenging for numerical studies Jaschke et al. (2017), it serves as a venue in which the quantum-classical hybrid implementation of QAOA could provide valuable input. Moreover, the small window is precisely the regime in which the QAOA approach requires only short-depth circuits.

Discussion and conclusion. — We have shown that QAOA-type protocols with long-range interactions allow for ultra-fast state preparation. In particular, the Ising-symmetric GHZ state can be prepared exactly at finite depth for odd (even) system sizes. We have also demonstrated that other states of interest, for example the quantum critical point of the Lipkin-Meshkov-Glick model, can also be prepared very efficiently. QAOA with long range interactions thus provides an opportunity for near-term simulators to prepare non-trivial states with very high fidelity and to shed light on areas of phase diagrams that are challenging for numerics.

Note added: While completing this manuscript, we became aware of related works Kokail et al. (2018) and et.al. ().

Acknowledgements.
Acknowledgments. – We thank R. Islam, C. Monroe, and B. Yoshida for useful discussions. WWH is supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through Grant No. GBMF4306. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

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Appendix A Supplemental Information

a.1 LMG Cost Function for

We evaluate

(9)

where

(10)

The second piece gives

(11)
(12)

Because any operator aside from identity and has zero expectation value in , we get contributions only from for each . In total, this piece is .

The first piece is

(13)
(14)

Again, we need only consider when the identity and operators arise. One contribution to the matrix element comes from the evolution of , which gives

(15)
(16)
(17)

Another contribution comes from

(18)
(19)

The transformation into two operators requires an odd number of applications of and ; each application comes with a factor of . The terms which do not alter the operator come with factors of . Hence, to single out the odd powers, we take the combination

(20)
(21)

In total, the cost function is thus

(22)

a.2 GHZ Preparation for Even

We show below that for an even number of qubits,

(23)

It is sufficient to establish

(24)

up to a phase. (The Ising symmetry operator is conserved as , so the matrix element for will also be ).

We break the matrix element in half and first the evaluate the left hand side. First,

(25)
(26)

where labels a spin configuration.

Applying and neglecting overall phase then gives

(27)
(28)

where we have defined .

The right hand side is:

(29)
(30)
(31)

where .

Consider the contributions to the coefficient of a given spin configuration . Each contribution involves partitioning the spins into two sets and of sizes and respectively, and flipping the spins in set . The resulting coefficient from this given flip is

(32)

where .

We now show that this factor only depends on the parity of (and the particular configuration ) and once this is fixed, the factor is independent of the partition. The final phase factor above can be written as

(33)

where . Because is even, is even. If is even, the is even and thus the phase factor is 1. Moreover, it is straightforward to check that either changing the partition (keeping partition size fixed) or changing the partition size by 2 does not change the above phase. Hence, the case of odd can be reduced to choosing to be the first spin. The wavefunction becomes

Dropping overall phases, we get

(34)

The matrix element between left and right hand sides is thus

(35)

Due to the last piece, any configuration with does not contribute and the matrix element reduces to

(36)
(37)
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