Quantum computing in geophysics: algorithms, computational costs and future applications Shahpoor Moradi 1, Daniel Trad 1, and Kristopher A. Innanen 1 1University of Calgary, Department of Geoscience, Calgary, Canada SUMMARY than their best classical algorithms (Nielsen and Chuang, 2000). In Figure 1 we show an illustration of a quantum Accurate modeling of seismic wave propagation in the search algorithm, which can locate a particular item in a subsurface of the earth is essential for understanding database in very few passes, independently of the size of earthquake dynamics, characterizing seismic hazards on the dataset. In each iteration, without actually measuring global scales and hydrocarbon reservoir exploration and the physical state of each sample, the amplitude of the monitoring on local scales. These are among the most desired answer is amplified until the correct answer is challenging computational problems in geoscience. Despite achieved with an optimal probability. algorithmic advances and the increasingly powerful computational resources currently available, including fast CPUs, GPUs and large volumes of computer memory, there are still daunting computational challenges in simulating 3D seismic wave propagation in complex earth environments. Recent advances in quantum computing are suggestive that geoscience may soon begin to benefit from this promising field. For example, Finite Difference (FD) modeling is the most widely used method to simulate seismic wave propagation. In the frequency domain, FD methods reduce solutions of the wave equation into systems Figure 1. Illustration of amplitude amplification in the search of linear equations; such systems are just the type that algorithm. The size of the search space is 64 and the number of quantum algorithms may be capable of solving with iterations to achieve the correct answer with high probability is 6. exponential speedup, in comparison with classical On the other hand, there are significant obstacles in the algorithms. For the computational geophysicist, to prepare application of quantum computers to numerical to take advantage of these speed-ups, which could arrive in computations. Unlike classical computers, error correction as few as 5-10 years, the tasks at hand are (1) to become is challenging for quantum computers. First, qubits are familiar with the logic and concepts associated with physical objects in nanoscale such as atoms, trapped ions or quantum computing, and (2) to map our key computational photons. Interactions of quantum computer with the algorithms (e.g., frequency domain FD) to this domain. environment affect the computation. Consequently, the first step to attain an error-free quantum computation is the CONCEPTS IN QUANTUM COMPUTATION isolation of physical qubits from any unwanted interactions. Second, direct measurement of numerical errors is Unlike a classical bit, which exists either in 0 or 1 states, a problematic because the act of measuring the state of a quantum bit (qubit), the building block of quantum qubit collapses it to a classical state, destroying the information, can exist in a coherent superposition of two quantum correlation which is crucial for the desired distinct states. In the quantum regime, 0 and 1 are two speedup. Third, since qubits can exist as a superposition of orthonormal states represented by Έand |0˿ = ʚ1 0ʛ |1˿ = multiple states, in addition to bit flip error, there is another Έ. A qubit is represented by , ʚ0 1ʛ | ˿ = |0˿ + |1˿ non-classical source of error called phase error. As an where and are probability amplitudes. Measuring the example, a qubit over time may change qubit results in either or , with respective |0˿ + |1˿ |0˿ |1˿ to . The standard way to recover a qubit from ͦ ͦ ͦ ͦ |0˿ − |1˿ probabilities | | and | | satisfying | | + | | = 1. In a quantum errors is to utilize ancillary qubits called physical ) quantum computer, a system of ͢ qubits can exist in 2 qubits. The idea is to transfer errors from logical qubits to states simultaneously, whereas in a classical computer, a physical qubits, and then discard them without measuring system of ͢ bits can store and process only one of 2) states the logical qubits. Initially, the number of physical qubits to at a given time. For example, a two-qubit system is capable correct for arbitrary errors from a single qubit was nine of holding four two bits of information simultaneously; (Shor, 1995) and later seven (Steane, 1996). The smallest i.e. |00 ˿, |01 ˿, |10 ˿ and |11 ˿. This coherent superposition is number of physical qubits that physicists have yet proposed the essence of the supremacy that quantum computing is five (Laflamme, 1996). Also, to allow a universal set of offers over its classical counterpart. Quantum algorithms quantum gates to be used on the logical qubits, we need are unambiguous instructions, based on effects such as some extra resources called magic states. These states are quantum coherence and entanglement, that are able to solve problems faster and with less computational complexity prepared using additional qubits and injected to the system To be able to investigate quantum algorithms in the (O'Gorman, and Campbell, 2017). absence of a scalable quantum computer, it is necessary to For example, implementation of factorization with Shor’s simulate their behavior on classical computers. This allows algorithm for a 2000-bit number (600 decimal digits) us to connect the theoretical advancement of quantum requires approximately 4000 computational logical qubits. computation with its physical realizations. We emphasize To successfully execute the algorithm with low error rate that no speedup should be expected when we run a we need about 3600 physical qubits (including the magic quantum algorithm on a classical computer; the exercise qubits) per logical qubit. In total, to run the Shor algorithm allows algorithms to be prepared for the introduction of a to factorize the 2000-digit number we need approximately real quantum computer. The largest classical simulation so one million qubits (Fowler, 2012). Evaluation time for this far was made in a 45-qubit circuit on the Cori II task with 1MHz speed takes four days. So far, the largest supercomputer using 8, 192 nodes and 0.5 petabytes of number that can be factorized on the most powerful memory (Häner, T. and Steiger, 2017). To put things in supercomputers is 768-bit number, and it took two years. perspective, the number of states that can be stored in 50 qubits is 2ͩͤ ~10 ͥͩ =1 Petabyte. There are various schemes for physical implementation of quantum computation including, atoms, photons, trapped Consequently, a 50-qubit simulation required over 1 ions and electrons (Nakahara and Ohmi, 2008). Other Petabyte of main memory to store results. potential candidates for qubits are superconducting circuits that behave as artificial atoms exhibiting a tunable energy. COMPUTATION CHALLENGES IN GEOSCIENCE States ‘0’ or ‘1’ correspond to the absence or presence of microwave photons or the oscillating electrical current Is it possible that a scalable quantum computer capable of through a loop. One of the important requirements to modeling 3D seismic wave propagation in a complex implement a quantum computer is to have long relevant model of the earth could be developed in the next few decoherence times, much longer than the gate operation years? To simulate seismic wave propagation in a 3D earth time. In other words, the lifetime of a qubit should be much model, especially for resolving the high-frequency content longer than the logic gate operation time. In this way, a of the data to capture small-scale features, we need a quantum computer has enough time to execute gates before supercomputer with peta (10 ͥͩ Flops) - or even exascale qubits’ coherence is destroyed. For example, for ion traps (10 ͥͬ Flops) power. For example to simulate wave decoherence time is 20s, which is much longer than gate modeling with a 10 ͬ grid cells and 10 ͥͤ degrees of operation times (0.01 ms). Electron spin qubits have the freedom we need a 1 PFlops supercomputer (Breuer, et al, fastest gate times (~ns) with fastest decoherence time 2014). As an example of the need for Tera-scale (~30 μs). On the hardware side, many research groups and computational power, the simulation of an M8 earthquake companies around the world are working on developing in the Southern California fault up to 2 Hz for 360s requires mini quantum computers for both commercial and research 220 TFlops on NCCS Jaguar using 223,074 cores for 24 purposes. Among them, D-wave is the sole company that hours. The simulation volume is 3.2 × 10 ͩkm ͦ with a offers a quantum simulator capable to solve certain tasks depth of 85 km meshed by 436 billion of 40 mͧ cubes (Cui known as optimization problems. Although D-wave is not a et al. 2010). Analytical solution of wave propagation in the universal quantum computer, it has been utilized for certain earth is possible with the assumption of spherical quantum calculations. The regime in which D-wave symmetry. However, a realistic model of the earth should operates is called quantum annealing. include ellipticity as well as heterogeneous structures in the Despite advances in the theoretical framework, the physical crust and mantle. For this reason, various numerical realization of quantum computing is still technologically modeling schemes used to simulate wave propagation in daunting. Reliable quantum computers that actually realistic 3D earth models, such as the forward modeling for demonstrate the exponential speed increase over their global and continental scales, are based on the Spectral classical counterparts are not going to be immediately Element Method (SEM). For local-scale exploration available. Well-known algorithms such as database search seismology, on the other hand, they are based on the finite- and prime factors of large integers have been demonstrated difference (FDM) or finite-element methods (FEM). on few qubits in labs, and this represents serious progress in the field.