Transcript of the presentation Programmable Quantum Simulators: Theory and Practice, given at the NTT Upgrade 2020 Research Summit, September 29, 2020

Hello. My name is Isaac Chuang and I am on the faculty at MIT in electrical engineering and computer science and in physics. And it is a pleasure for me to be presenting at today’s NTT research symposium of 2020 to share a little bit with you about programmable quantum simulators, theory and practice.

The simulation of physical systems as described by their Hamiltonian is a fundamental problem which Richard Fineman identified early on as one of the most promising applications of a hypothetical quantum computer. The real world around us, especially at the molecular level is described by Hamiltonians, which captured the interaction of electrons and nuclei. What we desire to understand from Hamiltonian simulation is properties of complex molecules, such as this iron molybdenum cofactor, an important catalyst. We desire there are ground states, reaction rates, reaction dynamics, and other chemical properties, among many things. For a molecule of N atoms, a classical simulation must scale exponentially with N, but for a quantum simulation, there is a potential for this simulation to scale polynomially instead, and this would be a significant advantage if realizable.

So where are we today in realizing such a quantum advantage? Today I would like to share with you a story about two things in this quest. First, a theoretical optimal quantum simulation algorithm, which achieves the best possible runtime for a generic Hamiltonian. Second, let me share with you experimental results from a quantum simulation implemented using available quantum computing hardware today with a hardware-efficient model that goes beyond what is utilized by today’s algorithms. I will begin with the theoretically optimal quantum simulation algorithm. In principle, the goal of quantum simulation is to take a time-independent Hamiltonian H and solve Schrodinger’s equation, as given here. This problem is as hard as the hardest quantum computation. It is known as being BQP Complete. A simplification, which is physically reasonable and important in practice, is to assume that the Hamiltonian is a sum over terms which are local, for example, due to a lattice structure.

These local terms typically do not commute, but their locality means that each term is reasonably small. Therefore, as was first shown by Seth Lloyd in 1996, one way to compute the time evolution, that is, the exponentiation of H with time, is to use the lead-product formula, which involves a successive approximation by repetitive small time steps. The cost of this charterization procedure is a number of elementary steps, which scales quadratically with the time desired and inverse with the error desired for the simulation output. Here N is the number of local terms in the Hamiltonian, and T is the desired simulation time where Epsilon is the desired simulation error. Today, we know that for special systems and higher order expansions of this formula, a better result can be obtained, such as scaling as N squared, but asymptotically linear in time. This, however, is for a special case, the lattice Hamiltonians, and it would be desirable to scale generally with time T for an order T time simulation.

So how could such an optimal quantum simulation be constructed? An important ingredient is to transform the quantum simulation into a quantum walk. This was done over 12 years ago by Andrew Childs showing that for sparse Hamiltonians with around d non-zero entries per row, such as shown in this graphic here, one can do a quantum walk very much like a classical walk, but in a superposition of right and left, shown here in this quantum circuit, where the H stands for a Hadamard Gate in this particular circuit. The Hadamard turns the zero into a superposition of zero and one, which then activate the left and the right walk in superposition. The graph of the walk is defined by the Hamiltonian H. And in doing so, Childs and collaborators were able to show the walk produces a unitary transform, which goes as e to the minus arc co-sine of H times time (t).

So this comes close, but it still has this transcendental function of H, instead of just simply H. This can be fixed with some effort, which results in an algorithm, which scales approximately as Tau log one over Epsilon, where Tau is proportional to the sparsity of the Hamiltonian and the simulation time. But again, the scaling here is a multiplicative product rather than an additive one, an interesting insight into the dynamics of a cubit. The simplest component of a quantum computer provides a way to improve upon this. Single cubits evolve as rotations in a sphere. For example, here is shown a rotation operator, which rotates around the axis Phi in the X-Y plane by angle theta. If one plots the results of this rotation as a projection along the Z axis, the result is a cosine squared function that is well-known as a Rabi oscillation. On the other hand, if a cubit is rotated around multiple angles in the X-Y plane, say around the phi equals zero, phi equals 1.5, and phi equals zero access again, then the resulting response function looks like a flat top.

And in fact, generalizing this to five or more pulses gives not just flatter hops, but in fact, arbitrary functions such as the Chebyshev polynomial shown here, which gives transforms like the Boolean OR and majority functions. Remarkably, if one does rotations by angle theta about D different angles in the X-Y plane, the result is a response function, which is a polynomial of order D in cosine theta. Furthermore, as captured by this theorem, given a nearly arbitrary degree polynomial, there exists angles phi such that one can achieve the desired polynomial. This is the result that derives from the Remez exchange algorithm used in classical discreet time signal processing.

So how does this relate to quantum simulation? Well, recall that a quantum walk essentially embeds a Hamiltonian inside the unitary transform of a quantum circuit. This embedding generalized might be called cubitization and it involves the use of a cubit acting as a projector to control the application of H. If we generalize the quantum walk to include a rotation about axis phi in the X-Y plane, it turns out that one obtains a polynomial transform of H itself.

And this polynomial is the same as the polynomial in the quantum signal processing theorem. This is a remarkable result known as the quantum singular value transform theorem from Andreas Gilyen, Yuan Su, Guang Hao Low, and Nathan Wiebe, published last year. This provides a quantum simulation algorithm using quantum signal processing. For example, can start with the quantum walk result and then apply quantum signal processing to undo the arc cosine transformation and therefore obtain the ideal expected Hamiltonian evolution E to the minus i H t. The resulting algorithm causes a number of elementary steps, which scales as just the sum of the evolution time and the log of one over the error desired. This saturates the known lower bound and thus is the optimal quantum simulation algorithm. This table from a recent review article summarizes a comparison of the query complexities of the known major quantum simulation algorithms showing that the cubitization and quantum signal processing algorithm is indeed optimal.

Of course, this optimality is a theoretical result. What does one do in practice? Let me now share with you the story of a hardware-efficient realization of a quantum simulation on actual hardware. The promise of quantum computation traditionally rests on a circuit model, such as the one we just used with quantum circuits acting on cubits. In contrast, consider a real physical problem from quantum chemistry, finding the vibronic structure of a molecule. The starting point is the Born-Oppenheimer separation of the electronic and vibrational states. For example, two connected nuclei share a vibrational mode. The potential energy of this nonlinear spring may be modeled as a harmonic oscillator. Since the spring’s energy is determined by the electronic structure, when the molecule becomes electronically excited, this vibrational mode changes. One obtains a different frequency and different equilibrium positions for the nuclei. This corresponds to a change in the spring constant, as well as a displacement of the nuclear positions. And we may write down a full Hamiltonian for this system.

The interesting quantum chemistry question is known as the Franck-Condon problem. What is the probability of transition between the original ground state and a given vibrational state in the excited state spectrum of the molecule? The Franck-Condon factor, which gives this transition probability, is foundational to quantum chemistry and a very hard and generic question to answer, which may be amiable to solution on a quantum computer. In particular, a natural quantum computer to use might be one which already has harmonic oscillators, rather than one which has just cubits. This is provided in a Bosonic Quantum Processor, such as the superconducting cubits system shown here. This processor has both cubits as embodied by the Josephson-junction shown here, and a harmonic oscillator as embodied by the resonant mode of the transmission cavity given here. Moreover, the output of this planar superconducting circuit can be connected to three dimensional cavities. Instead of using cubit gates, one may perform direct transformations on the Bosonic state, using for example, beam splitters, phase shifters, displacement, and squeezing operators, and the harmonic oscillator may be initialized and manipulated directly. The availability of the cubit allows photon number resolve counting.

For simulating a tri-atomic, two-mode, Franck-Condon factor problem, this superconducting cubits system with 3D cavities was employed. Two resonators, cavity A and cavity B, represent the breathing and wiggling modes of a tri-atomic molecule, as depicted here. The coupling of these modes was mediated by a superconducting cubit and read-out was accomplished by two additional superconducting cubits coupled to each one of the cavities. Due to the superconducting resonators used, each one of the cavities had a long coherence time, while resonator states could be prepared and measured using these strong coupling of cubits to the cavity. And Bosonic quantum operations could be realized by modulating the coupling cubit in between the two cavities. The cavities are holes drilled into pure aluminum, kept superconducting by millikelvin-scale temperatures.

Microfiber[not clear] chips with superconducting cubits are inserted into ports to couple via an antenna to the microwave cavities. Each of the cavities has a quality factor so high that the coherence times can reach milliseconds. A coupling cubit chip is inserted into the port in between the cavities, and the readout and preparation cubit chips are inserted into ports on the sides. For sake of brevity, I will skip the experimental details and present just the results. Shown here is the vibrotic spectrum obtained for a water molecule using the Bosonic superconducting processor. This is a typical Franck-Condon spectrum giving the intensity of lines versus frequency in wave number, where the solid line depicts the theoretically expected result and the purple and red dots show two sets of experimental data: one taken quickly, and another taken with exhaustive statistics. In both cases, the experimental results have good agreement with the theoretical expectations. The programmability of this system is demonstrated by showing how it can easily calculate the Franck-Condon spectrum for a wide variety of molecules.

Here’s another one, the ozone anion. Again, we see that the experimental data shown in points agrees well with the theoretical expectation shown as a solid line. Let me emphasize that this quantum simulation result was obtained not by using a quantum computer with cubits, but rather one with resonators, one resonator representing each one of the modes of vibration in this tri-atomic molecule. This approach represents a far more efficient utilization of hardware resources compared with the standard cubit model because of the natural match of the resonators with the physical system being simulated. In comparison, if cubit gates had been utilized to perform the same simulation, on the order of a thousand cubit gates would have been required compared with the order of 10 operations, which were performed for this Bosonic realization.

Asymptotically, the cubit model would have required significantly more operations because of the need to discretize each one of the harmonic oscillators into some max Hilbert-space size.  Compared with the optimal quantum simulation algorithms shown in the first half of this talk, we see that there is a significant gap between what available quantum computing hardware can perform and what optimal quantum simulations demand in terms of the number of gates required for a simulation. Nevertheless, many of the techniques that are used for optimal quantum simulation algorithms may become useful, especially if they are adapted to available hardware. Moving forward, the future holds some interesting challenges for this field.

Real physical systems are not cubits. Rather, they are composed from bosons and fermions. And fermions need global antisymmetrization. This is a huge challenge for electronic structure calculation in molecules. Real physical systems also have symmetries, but current quantum simulation algorithms are largely governed by a theorem, which says that the number of times steps required is proportional to the simulation time desired. Finally, real physical systems are not purely quantum or purely classical, but rather have many messy quantum-classical boundaries. In fact, perhaps the most important systems to simulate are really open quantum systems. And these dynamics are described by a mixture of quantum and classical evolution, and the desired results are often thermal and statistical properties.

I hope this presentation of the theory and practice of quantum simulation has been interesting and worthwhile. Thank you.