Article

Bravyi-Kitaev Superfast simulation of fermions on a quantum computer

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

Present quantum computers often work with distinguishable qubits as their computational units. In order to simulate indistinguishable fermionic particles, it is first required to map the fermionic state to the state of the qubits. The Bravyi-Kitaev Superfast (BKSF) algorithm can be used to accomplish this mapping. The BKSF mapping has connections to quantum error correction and opens the door to new ways of understanding fermionic simulation in a topological context. Here, we present the first detailed exposition of BKSF algorithm for molecular simulation. We provide the BKSF transformed qubit operators and report on our implementation of the BKSF fermion-to-qubits transform in OpenFermion. In this initial study of the hydrogen molecule, we have compared BKSF, Jordan-Wigner and Bravyi-Kitaev transforms under the Trotter approximation. We considered different orderings of the exponentiated terms and found lower Trotter errors than previously reported for Jordan-Wigner and Bravyi-Kitaev algorithms. These results open the door to further study of the BKSF algorithm for quantum simulation.

No full-text available

... Generalization of the transformation to higher dimensions using lattice gauge fields have also been proposed [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Such exact bosonization mappings are also important form the quantum simulation perspective, for they address the possibility of simulating a quantum many-body fermionic systems using a bosonic quantum computer [21][22][23][24][25][26]. Symmetries, however, may not remain manifest under the transformation. ...
... The ηχ approach is inspired by earlier works provid-ing lattice bosonization recipes through the introduction of Majorana fermions [7,8,23]. It is also closely related to the approach developed in Refs. ...
Preprint
Full-text available
We discuss a scheme for performing Jordan-Wigner transformation for various lattice fermion systems in two and three dimensions which keeps internal and spatial symmetries manifest. The correspondence between fermionic and bosonic operators is established with the help of auxiliary Majorana fermions. The current construction is applicable to general lattices with even coordination number and an arbitrary number of fermion flavors. The approach is demonstrated on the single-orbital square, triangular and cubic lattices for spin-1/2 fermions. We also discuss the relation to some quantum spin liquid models.
... However, there is a mapping with which locally-interacting Fermion and qubit lattices can be related: the Verstraete-Cirac transform (VCT) [23] also known as Auxiliary Fermion Mapping [24][25][26], can be regarded as a manipulation of the Jordan-Wigner transform, in which additional auxiliary particles are added, hence the name. Other works on Fermion-to-qubit mappings [25,[27][28][29] are based on two transforms proposed by Bravyi and Kitaev in [30]. Firstly, there is what is commonly known as the Bravyi-Kitaev transform, that, compared to the Jordan-Wigner transform, exhibits an up to exponential improvement on the number of qubits that each fermionic interaction term acts on. ...
... So far, we still have not left the even-parity subspace, but we might have systems to solve that are populated by odd numbers of Fermions. In [29], it is suggested to add another mode to the system that is however not coupled to any other term in the Hamiltonian. From the original concept of the BKSF it is however not clear how this mode is brought into the system, since all qubits correspond to couplings of modes in the Hamiltonian, which here do not exist. ...
Preprint
Full-text available
Quantum simulation of fermionic systems is a promising application of quantum computers, but in order to program them, we need to map fermionic states and operators to qubit states and quantum gates. While quantum processors may be build as two-dimensional qubit networks with couplings between nearest neighbors, standard Fermion-to-qubit mappings do not account for that kind of connectivity. In this work we concatenate the (one-dimensional) Jordan-Wigner transform with specific quantum codes defined under the addition of a certain number of auxiliary qubits. This yields a novel class of mappings with which any fermionic system can be embedded in a two-dimensional qubit setup, fostering scalable quantum simulation. Our technique is demonstrated on the two-dimensional Fermi-Hubbard model, that we transform into a local Hamiltonian. What is more, we adapt the Verstraete-Cirac transform and Bravyi-Kitaev Superfast simulation to the square lattice connectivity and compare them to our mappings. An advantage of our approach in this comparison is that it allows us to encode and decode a logical state with a simple unitary quantum circuit.
... Constructing all generators directly in their fermionic representation, will result in automatic gradient evaluation schemes independent of the underlying qubit encoding, allowing flexible adaption of new encodings [44,45] and improved compiling strategies into quantum gates [46]. In analogy to the qubit construction in Eq. (29) the nullspace projector of a general single, double or n-fold fermionic excitation (5) can be constructed as ...
Preprint
We develop computationally affordable and encoding independent gradient evaluation procedures for unitary coupled-cluster type operators, applicable on quantum computers. We show that, within our framework, the gradient of an expectation value with respect to a parameterized n-fold fermionic excitation can be evaluated by four expectation values of similar form and size, whereas most standard approaches based on the direct application of the parameter-shift-rule come with an associated cost of O(2^(2n)) expectation values. For real wavefunctions, this cost can be further reduced to two expectation values. Our strategies are implemented within the open-source package tequila and allow blackboard style construction of differentiable objective functions. We illustrate initial applications for electronic ground and excited states.
Article
Full-text available
Trotter–Suzuki decompositions are frequently used in the quantum simulation of quantum chemistry. They transform the evolution operator into a form implementable on a quantum device, while incurring an error—the Trotter error. The Trotter error can be made arbitrarily small by increasing the Trotter number. However, this increases the length of the quantum circuits required, which may be impractical. It is therefore desirable to find methods of reducing the Trotter error through alternate means. The Trotter error is dependent on the order in which individual term unitaries are applied. Due to the factorial growth in the number of possible orderings with respect to the number of terms, finding an optimal strategy for ordering Trotter sequences is difficult. In this paper, we propose three ordering strategies, and assess their impact on the Trotter error incurred. Initially, we exhaustively examine the possible orderings for molecular hydrogen in a STO-3G basis. We demonstrate how the optimal ordering scheme depends on the compatibility graph of the Hamiltonian, and show how it varies with increasing bond length. We then use 44 molecular Hamiltonians to evaluate two strategies based on coloring their incompatibility graphs, while considering the properties of the obtained colorings. We find that the Trotter error for most systems involving heavy atoms, using a reference magnitude ordering, is less than 1 kcal/mol. Relative to this, the difference between ordering schemes can be substantial, being approximately on the order of millihartrees. The coloring-based ordering schemes are reasonably promising—particularly for systems involving heavy atoms—however further work is required to increase dependence on the magnitude of terms. Finally, we consider ordering strategies based on the norm of the Trotter error operator, including an iterative method for generating the new error operator terms added upon insertion of a term into an ordered Hamiltonian.
Article
Full-text available
We present a quantum algorithm for simulating quantum chemistry with gate complexity $$\tilde {\cal{O}}(N^{1/3}\eta ^{8/3})$$ where η is the number of electrons and N is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity $$\tilde {\cal{O}}(N^{8/3}{\mathrm{/}}\eta ^{2/3})$$. We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when N ≫ η, as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation.
Article
Full-text available
Current implementations of the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem involve splitting the system qubit Hamiltonian into parts whose elements commute within their single qubit subspaces. The number of such parts rapidly grows with the size of the molecule. This increases the computational cost and can increase uncertainty in the measurement of the energy expectation value because elements from different parts need to be measured independently. To address this problem we introduce a more efficient partitioning of the qubit Hamiltonian using fewer parts that need to be measured separately. The new partitioning scheme is based on two ideas: 1) grouping terms into parts whose eigenstates have a single-qubit product structure, and 2) devising multi-qubit unitary transformations for the Hamiltonian or its parts to produce less entangled operators. The first condition allows the new parts to be measured in the number of involved qubit consequential one-particle measurements. Advantages of the new partitioning scheme resulting in severalfold reduction of separately measured terms are illustrated on the H2 and LiH problems.
Article
The ability to perform classically intractable electronic structure calculations is often cited as one of the principal applications of quantum computing. A great deal of theoretical algorithmic development has been performed in support of this goal. Most techniques require a scheme for mapping electronic states and operations to states of and operations upon qubits. The two most commonly used techniques for this are the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. However, comparisons of these schemes have previously been limited to individual small molecules. In this paper we discuss resource implications for the use of the Bravyi-Kitaev mapping scheme, specifically with regard to the number of quantum gates required for simulation. We consider both small systems which may be simulatable on near-future quantum devices, and systems sufficiently large for classical simulation to be intractable. We use 86 molecular systems to demonstrate that the use of the Bravyi-Kitaev transformation is typically at least approximately as efficient as the canonical Jordan-Wigner transformation, and results in substantially reduced gate count estimates when performing limited circuit optimisations.
Article
Full-text available
We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure.
Article
Quantum computers can be used to address electronic-structure problems and problems in materials science and condensed matter physics that can be formulated as interacting fermionic problems, problems which stretch the limits of existing high-performance computers. Finding exact solutions to such problems numerically has a computational cost that scales exponentially with the size of the system, and Monte Carlo methods are unsuitable owing to the fermionic sign problem. These limitations of classical computational methods have made solving even few-atom electronic-structure problems interesting for implementation using medium-sized quantum computers. Yet experimental implementations have so far been restricted to molecules involving only hydrogen and helium. Here we demonstrate the experimental optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms, determining the ground-state energy for molecules of increasing size, up to BeH2. We achieve this result by using a variational quantum eigenvalue solver (eigensolver) with efficiently prepared trial states that are tailored specifically to the interactions that are available in our quantum processor, combined with a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine. We demonstrate the flexibility of our approach by applying it to a problem of quantum magnetism, an antiferromagnetic Heisenberg model in an external magnetic field. In all cases, we find agreement between our experiments and numerical simulations using a model of the device with noise. Our results help to elucidate the requirements for scaling the method to larger systems and for bridging the gap between key problems in high-performance computing and their implementation on quantum hardware.
Article
It is shown that an arbitrary fermion hopping Hamiltonian can be mapped into a system with no fermion fields, generalizing an earlier model of Levin and Wen. All operators in the Hamiltonian of the resulting description commute (rather than anticommute) when acting at different sites, despite the system having excitations obeying Fermi statistics. While extra conserved degrees of freedom are introduced, they are all locally identified in the representation obtained. The same methods apply to Majorana (half) fermions, which for Cartesian lattices mitigate the fermion doubling problem. The generality of these results suggests that the observation of Fermion excitations in nature does not demand that anticommuting Fermion fields be fundamental.
Article
INTRODUCTION E-commerce and e-business on the Internet with worldwide round-the-clock operation require enterprises to be operational 24 hours per day, 7 days per week. Outages are very expensive for these enterprises, with accompanying loss of revenue and reputation, and with disgruntled customers. Table I gives the estimates of the financial losses [1] due to downtime for different types of enterprises. With an increasing demand for 24 7 operation, and with downtime being unacceptable and prohibitive in cost, enterprises must have fault tolerance to survive in today's marketplace. # Correspondence to: P. Narasimhan, School of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A. + E-mail: priya@cs.cmu.edu Contract/grant sponsor: Defense Advanced Research Projects Agency in conjunction with the office of Naval Research and the Air Force Research Laboratory; contract/grant number: N00174-95-K-0083, F3602-97-1-0248. Copyright # 2002 John Wiley & Sons, Ltd.
• M Mohseni
• P Reed
• H Neven
M. Mohseni, P. Reed, and H. Neven, Nature 543 (2017).
• P J J O'malley
• R Babbush
• I D Kivlichan
• J Romero
• J R Mcclean
• R Barends
• J Kelly
• P Roushan
• A Tranter
• N Ding
P. J. J. O'Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, et al., Physical Review X 6, 031007 (2016).
• R Barends
• J Kelly
• A Megrant
• A Veitia
• D Sank
• E Jeffrey
• T C White
• J Mutus
• A G Fowler
• B Campbell
R. Barends, J. Kelly, A. Megrant, A. Veitia, D. Sank, E. Jeffrey, T. C. White, J. Mutus, A. G. Fowler, B. Campbell, et al., Nature 5, 4213 (2014).
• J Kelly
• R Barends
• B Campbell
• Y Chen
• Z Chen
• B Chiaro
• A Dunsworth
• A G Fowler
• I.-C Hoi
• E Jeffrey
J. Kelly, R. Barends, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, I.-C. Hoi, E. Jeffrey, et al., Physical Review Letters 112, 240504 (2014).
• A Peruzzo
• J Mcclean
• M.-H Yung
• X.-Q Zhou
• P J Love
• J L O Brien
A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, and J. L. O. Brien, Nature Communications 5, 4213 (2014).
• J Du
• N Xu
• X Peng
• P Wang
• S Wu
• D Lu
J. Du, N. Xu, X. Peng, P. Wang, S. Wu, and D. Lu, Physical Review Letters 104, 030502 (2010).
• S P Jordan
• K S M Lee
• J Preskill
S. P. Jordan, K. S. M. Lee, and J. Preskill, Science 336, 1130 (2012).
• E Zohar
• A Farace
• B Reznik
• J I Cirac
E. Zohar, A. Farace, B. Reznik, and J. I. Cirac, Physical Review Letters 118 (2017).
• B P Lanyon
• J D Whitfield
• G G Gillett
• M E Goggin
• M P Almeida
• I Kassal
• J D Biamonte
• M Mohseni
• B J Powell
• M Barbieri
B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, et al., Nature Chemistry 2, 106 (2010).
• M.-H Yung
• J D Whitfield
• S Boixo
• D G Tempel
• A Aspuru-Guzik
M.-H. Yung, J. D. Whitfield, S. Boixo, D. G. Tempel, and A. Aspuru-Guzik, Introduction to Quantum Algorithms for Physics and Chemistry (John Wiley & Sons, Inc., 2014).
• G Ortiz
• J E Gubernatis
• E Knill
• R Laflamme
G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Physical Review A 64, 022319 (2001).
• V Havlíček
• M Troyer
• J D Whitfield
V. Havlíček, M. Troyer, and J. D. Whitfield, Physical Review A 95, 032332 (2017).
• J T Seeley
• M J Richard
• P J Love
J. T. Seeley, M. J. Richard, and P. J. Love, The Journal of Chemical Physics 137, 224109 (2012).
• S B Bravyi
• A Yu Kitaev
S. B. Bravyi and A. Yu Kitaev, Annals of Physics 298, 210 (2000).
• F Verstraete
• J I Cirac
F. Verstraete and J. I. Cirac, Journal of Statistical Mechanics: Theory and Experiment 09, P09012 (2005).
• C Zalka
C. Zalka, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 454, 313 (1998).
• D A Lidar
• H Wang
D. A. Lidar and H. Wang, Physical Review E 59, 2429 (1999).
• S Lloyd
S. Lloyd, Science 273, 1073 (1996).
• L Veis
• J Višňák
• T Fleig
• S Knecht
• T Saue
• L Visscher
• J C V Pittner
L. Veis, J. Višňák, T. Fleig, S. Knecht, T. Saue, L. Visscher, and J. c. v. Pittner, Physical Review A 85, 030304 (2012).
• D W Berry
• A M Childs
• R Cleve
• R Kothari
• R D Somma
D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, and R. D. Somma, Physical Review Letters 114, 090502 (2015).
• J R Mcclean
• J Romero
• R Babbush
• A Aspuru-Guzik
J. R. McClean, J. Romero, R. Babbush, and A. Aspuru-Guzik, New Journal of Physics 18, 023023 (2016).
• S Paesani
• A A Gentile
• R Santagati
• J Wang
• N Wiebe
• D P Tew
• J L O'brien
• M G Thompson
S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. O'Brien, and M. G. Thompson, Physical Review Letters 118, 100503 (2017).
• I Chuang
• M Nielsen
I. Chuang and M. Nielsen, Quantum Computation and Quantum Information (Cambridge University Press, 2000).
• A Y Kitaev
• A H Shen
• M N Vyalyi
A. Y. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and Quantum Computation (American Mathematical Society, 2002).
• R Somma
• G Ortiz
• J E Gubernatis
• E Knill
• R Laflamme
R. Somma, G. Ortiz, J. E. Gubernatis, E. Knill, and R. Laflamme, Physical Review A 65, 042323 (2002).
• A Aspuru-Guzik
• A D Dutoi
• P J Love
A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, Science 309, 1704 (2005).
• J D Whitfield
• J Biamonte
• A Aspuru-Guzik
J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Molecular Physics 109, 735 (2011).
• L Veis
• J Pittner
L. Veis and J. Pittner, The Journal of Chemical Physics 133, 194106 (2010).
• H Wang
• S Kais
• A Aspuru-Guzik
• M R Hoffmann
H. Wang, S. Kais, A. Aspuru-Guzik, and M. R. Hoffmann, Physical Chemistry Chemical Physi 10, 5388 (2008).
• R M Parrish
• L A Burns
• D G A Smith
• A C Simmonett
• A E Deprince
• E G Hohenstein
• U Bozkaya
• A Y Sokolov
• R Di Remigio
• R M Richard
R. M. Parrish, L. A. Burns, D. G. A. Smith, A. C. Simmonett, A. E. DePrince, E. G. Hohenstein, U. Bozkaya, A. Y. Sokolov, R. Di Remigio, R. M. Richard, et al., Journal of Chemical Theory and Computation 13, 3185 (2017).
• M B Hastings
• D Wecker
• B Bauer
• M Troyer
M. B. Hastings, D. Wecker, B. Bauer, and M. Troyer, Quantum Information & Computation 15, 1 (2015).
• A Kitaev
A. Kitaev, Annals of Physics 321, 2 (2006).
• J K Pachos
J. K. Pachos, Annals of Physics 322, 1254 (2007).
• A Y Kitaev
A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001).
• K Paton
K. Paton, Communications of the ACM 12, 514 (1969).
• A Szabo
• N S Ostlund
A. Szabo and N. S. Ostlund, Modern Quatnum Chemistry (Dover Publications, 1996).
• J B Collins
• P V R Schleyer
• J S Binkley
• J A Pople
J. B. Collins, P. v. R. Schleyer, J. S. Binkley, and J. A. Pople, Journal of Chemical Physics 76, 5142 (1976).
• A M Childs
• T Li
A. M. Childs and T. Li (2016), arXiv:1611.05543.
• J Romero
• R Babbush
• J R Mcclean
• C Hempel
• P Love
J. Romero, R. Babbush, J. R. McClean, C. Hempel, and P. Love (2017), arXiv: 1701.02691.
• J R Mcclean
• I D Kivlichan
• D S Steiger
• Y Cao
• E S Fried
• C Gidney
• T Haner
• V Havlicek
• Z Jiang
• M Neeley
J. R. McClean, I. D. Kivlichan, D. S. Steiger, Y. Cao, E. S. Fried, C. Gidney, T. Haner, V. Havlicek, Z. Jiang, M. Neeley, et al. (2017), arXiv:1710.07629.
• S Bravyi
• J M Gambetta
• A Mezzacapo
• K Temme
S. Bravyi, J. M. Gambetta, A. Mezzacapo, and K. Temme (2017), arXiv:1701.08213.
• P Jordan
• E Wigner
P. Jordan and E. Wigner, Zeitshrift für Physik 47, 631 (1928).
• S Wiesner
S. Wiesner (1996), arXiv:quant-ph/9603028.
• I Kassal
• S P Jordan
• P J Love
• M Mohseni
• A Aspuru-Guzik
I. Kassal, S. P. Jordan, P. J. Love, M. Mohseni, and A. Aspuru-Guzik, Proceedings of the National Academy of Sciences 105, 18681 (2008).
• M Suzuki
M. Suzuki, Physical Letters A 165, 387 (1992).
• D S Steiger
• T Haner
• M Troyer
D. S. Steiger, T. Haner, and M. Troyer (2016), arXiv:1612.08091.