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The Bravyi-Kitaev transformation for quantum computation of electronic structure

Jacob T. Seeley, Martin J. Richard, Peter J. Love

arXiv:1208.5986v1quant-ph

TL;DR

The paper applies the Bravyi-Kitaev transformation to fermionic operators in quantum chemical Hamiltonians. For H2 in a minimal basis, the resulting simulation circuit uses fewer gates than the Jordan-Wigner version, supporting the transformation’s superior efficiency for quantum chemical simulations.

  • Problem

    Quantum simulation of electronic structure requires a representation of fermions.

  • Method

    The paper works out a detailed application of the Bravyi-Kitaev transformation to Hermitian second-quantized operators in quantum chemical Hamiltonians.

  • Results

    328 gates versus 222 gates are required for electron-dynamics simulation of H2 in a minimal basis.

  • Takeaways & Limitations

    The authors conclude that Bravyi-Kitaev has superior efficiency for quantum chemical simulations and should replace Jordan-Wigner.

Abstract

from arXiv · show

Quantum simulation is an important application of future quantum computers with applications in quantum chemistry, condensed matter, and beyond. Quantum simulation of fermionic systems presents a specific challenge. The Jordan-Wigner transformation allows for representation of a fermionic operator by O(n) qubit operations. Here we develop an alternative method of simulating fermions with qubits, first proposed by Bravyi and Kitaev [S. B. Bravyi, A.Yu. Kitaev, Annals of Physics 298, 210-226 (2002)], that reduces the simulation cost to O(log n) qubit operations for one fermionic operation. We apply this new Bravyi-Kitaev transformation to the task of simulating quantum chemical Hamiltonians, and give a detailed example for the simplest possible case of molecular hydrogen in a minimal basis. We show that the quantum circuit for simulating a single Trotter time-step of the Bravyi-Kitaev derived Hamiltonian for H2 requires fewer gate applications than the equivalent circuit derived from the Jordan-Wigner transformation. Since the scaling of the Bravyi-Kitaev method is asymptotically better than the Jordan-Wigner method, this result for molecular hydrogen in a minimal basis demonstrates the superior efficiency of the Bravyi-Kitaev method for all quantum computations of electronic structure.

I. INTRODUCTION

Quantum simulation of fermionic systems requires efficient mappings from fermionic operators and electronic Hamiltonians to qubit operations. This paper develops and applies the Bravyi-Kitaev transformation, showing lower resource requirements than Jordan-Wigner for molecular hydrogen.

  • Motivation: Quantum simulation of electronic structure requires a qubit representation of fermions, while gate-count scaling remains challenging.The introduction frames efficient simulation as important for quantum chemistry and identifies the scaling of existing algorithms as a continuing problem.
  • Contribution: The Bravyi-Kitaev basis and transformation provide a more efficient mapping between electronic Hamiltonians and qubit Hamiltonians.The paper develops the transformation specifically for operators appearing in quantum chemical Hamiltonians.
  • Contribution: O(log n) qubit operations represent one fermionic creation or annihilation operator with Bravyi-Kitaev, compared with O(n) for Jordan-Wigner.This reduction is the central asymptotic advantage claimed for the new representation.
  • Application: For H2 in a minimal basis, the paper gives explicit Pauli decompositions and compares first-order Trotter circuits derived from both transformations.The example targets the simplest molecular system and makes the resource comparison concrete.
  • Results: 30 single-qubit gates and 44 CNOT gates are required for one Bravyi-Kitaev Trotter time-step, versus 46 single-qubit gates and 36 CNOT gates for Jordan-Wigner.The Bravyi-Kitaev circuit uses fewer total gates despite using more CNOT gates in this example.
  • Results: 222 total gates across 3 first-order Trotter steps achieve chemical precision, compared with 328 gates across 4 steps for Jordan-Wigner.The paper uses this H2 result together with the asymptotic scaling to argue for superior Bravyi-Kitaev efficiency in molecular simulations.

II. BACKGROUND

Electronic structure is formulated using fermionic basis states and creation and annihilation operators. The molecular electronic Hamiltonian is expressed through one- and two-electron integrals, with molecular hydrogen providing a four-spin-orbital example.

  • Fermionic systems and second quantization: Second quantization describes fermionic systems using occupation-number basis states in which each orbital is empty or occupied.Occupation numbers are restricted to 0 and 1 by the Pauli exclusion principle.
  • Fermionic systems and second quantization: Creation and annihilation operators change orbital occupations and acquire phases determined by fermionic exchange antisymmetry.For an orbital j, the phase depends on occupations of orbitals with lower index under the chosen ordering.
  • Fermionic systems and second quantization: The canonical fermionic anticommutation relations encode the required exchange antisymmetry of these operators.The anticommutator is defined as [A, B]+ ≡ AB + BA.
  • Electronic Hamiltonian: The molecular electronic Hamiltonian is built from one- and two-electron overlap integrals that can be precomputed classically.These integrals are supplied to the quantum simulation as parameters.
  • Molecular hydrogen example: In the minimal-basis H2 model, two spatial molecular orbitals combine with two spin functions to form four molecular spin orbitals.The spatial orbitals are gerade and ungerade combinations, and each is paired with α or β spin.

B. The Jordan-Wigner transformation

The Jordan-Wigner transformation maps occupation numbers directly to qubits and uses Pauli-z strings to reproduce fermionic phases and anticommutation. Its extra-operation cost for one fermionic operator scales as O(n).

  • Occupation-number encoding: The occupation-number encoding stores the occupation f_j of orbital j in qubit j.This direct mapping identifies each fermionic basis bit with the corresponding qubit state.
  • Operator mapping: A direct tensor-product construction of qubit creation and annihilation operators fails because the resulting operators do not obey fermionic anticommutation relations.The Jordan-Wigner σz string corrects this mismatch by reproducing the required phases.
  • Operator mapping: Jordan-Wigner represents fermionic creation and annihilation operators using one-qubit operators together with σz operations on lower-index qubits.The σz string supplies the phase determined by the parity of preceding occupations.
  • Operator mapping: The parity of lower-index qubits produces phase −1 for odd parity and no phase change for even parity.Parity is the sum modulo 2 of the qubit-state numbers in the relevant set.
  • Cost: O(n) extra qubit operations are required to simulate one fermionic operator with Jordan-Wigner.The linear overhead arises from the string of operations extending over lower-index qubits.
  • Alternative basis: The parity-basis alternative still has an O(n) leading σx update string, so it does not improve Jordan-Wigner efficiency.The paper introduces the Bravyi-Kitaev construction as a third possibility after identifying this limitation.

B. The Bravyi-Kitaev basis

The Bravyi-Kitaev basis balances occupation and parity information by storing parities of structured orbital subsets, reducing the information each fermionic operation must access.

  • Qubits store the parity of orbital sets, with even-indexed qubits storing one orbital and odd-indexed qubits also storing selected adjacent lower-indexed orbitals.
  • The encoding maps occupation-number bit strings to Bravyi-Kitaev bit strings through a binary-structured matrix βn, with additions performed modulo 2.
  • Compared with occupation-number and parity encodings, Bravyi-Kitaev stores occupation and parity information in a middle ground that improves locality for both.
  • O(log n) partial sums suffice for the relevant parity information, and only qubits storing partial sums that include orbital j require updates.
  • The encoding lets one identify parity, update, and flip qubit sets needed to construct fermionic creation and annihilation operators.

IV. SETS OF QUBITS RELEVANT TO THE BRAVYI-KITAEV BASIS

The Bravyi-Kitaev transformation organizes three qubit subsets that determine parity, update stored partial sums, and resolve the parity relation of the target qubit.

  • The parity set P(j) contains qubits whose parity matches all orbitals with index less than j.
  • The update set contains qubits storing partial sums that include orbital j.
  • The flip set contains qubits that determine whether qubit j has the same or inverted parity relative to orbital j.
  • The transformation matrix and its inverse identify these sets through nonzero entries positioned relative to the main diagonal.
  • The parity set determines whether a creation or annihilation operation on orbital j introduces a phase change of −1.

B. The update set

The update set identifies which Bravyi-Kitaev qubits must change when orbital j changes, and its logarithmic structure limits the update cost.

  • U(j) contains Bravyi-Kitaev qubits storing partial sums that include occupation number j.
  • Even-indexed qubits store only their corresponding orbital, so update sets contain only odd indices.
  • The transformation matrix βn reveals update-set elements through nonzero entries in the relevant column above the main diagonal.
  • Update sets depend on basis size; with 16 basis functions instead of 8, every update set except U(7) also includes index 15.
  • The defined sets support derivation of the Bravyi-Kitaev mapping from fermionic operators to equivalent qubit operators.

V. THE BRAVYI-KITAEV TRANSFORMATION

The Bravyi-Kitaev transformation represents fermionic creation and annihilation operators with parity, update, and flip operations on logarithmically many qubits.

  • Fermionic operators are represented using parity projectors expressed through products of Pauli Z operators, together with qubit creation and annihilation operators.
  • Even-indexed orbitals: For even j, the construction applies Q± on qubit j, Z operations on P(j), and X operations on U(j).
  • Scaling: O(log n) qubits suffice for the parity and update sets, so one fermionic operator has simulation cost O(log n).
  • Odd-indexed orbitals: The parity of F(j) determines whether creation and annihilation use the usual or reversed qubit operators for odd j.
  • Unified representation: Odd-indexed operators use a remainder set R(j) in place of P(j) for the relevant Z term, yielding a unified arbitrary-j expression.
  • Operator products: Compact algebraic expressions are given for number, Coulomb, and exchange operators, while the remaining classes require general product expressions because explicit cases become impractical.

A. Number operators: hii a†

The paper derives Bravyi–Kitaev-basis expressions for number operators and related operator products by simplifying overlapping Pauli supports across index-parity cases.

  • A. Number operators: hii a†: Number operators have eigenvalues corresponding to orbital occupation numbers and are represented in the Bravyi–Kitaev basis.The derivation treats arbitrary orbital indices, including even and odd cases.
  • A. Number operators: hii a†: The Bravyi–Kitaev expressions exploit parity-dependent update and parity sets to simplify local Pauli products.For even indices, parity and remainder sets coincide, producing an empty symmetric difference; odd indices require the corresponding remainder set.
  • A. Number operators: hii a†: Coulomb and exchange operators can be treated as products of two number operators and reduced by simplifying overlapping supports.Overlapping Z operators cancel locally, so Z operations are needed only on the union of the relevant sets.
  • A. Number operators: hii a†: Products of excitation operators are simplified by classifying overlaps among update and parity sets, with local products yielding X, Y, or identity factors.The derivation assumes i < j and distinguishes cases according to the parity of the indices.
  • A. Number operators: hii a†: The general product formulas are summarized in Table II and used to generate excitation, number-excitation, and double-excitation operators.The derivation gives one case in detail and presents the remaining parity cases through the table.

D. Excitation operators: hij (a†

The paper constructs excitation and number-excitation operators in the Bravyi–Kitaev basis by combining parity-dependent Pauli expressions and simplifying their overlapping supports.

  • D. Excitation operators: hij (a†: Complex one-electron integrals are accommodated by separating real and imaginary parts in the excitation-operator expressions.For even indices, the resulting expression contains XjXi, YjYi, and a term proportional to the imaginary part of hij.
  • D. Excitation operators: hij (a†: Number-excitation operators are formed by combining an excitation operator with a number operator.The paper uses previously derived algebraic expressions for both operator classes.
  • D. Excitation operators: hij (a†: Simplification depends on intersections between the number operator’s support and the excitation operator’s support.The form changes with these set intersections, while the underlying reduction process remains the same.
  • D. Excitation operators: hij (a†: For non-even index combinations, the procedure changes primarily through the appropriate excitation-operator form from Table II.The paper notes that the local-support simplification itself is unchanged.
  • D. Excitation operators: hij (a†: Double-excitation operators are the most algebraically complicated class because four distinct indices create many index-dependent subcases.The paper therefore outlines the derivation procedure rather than enumerating every subcase.

VII. THE MOLECULAR ELECTRONIC HAMILTONIAN FOR THE HYDROGEN MOLECULE IN THE BRAVYI-KITAEV BASIS

The paper applies the Bravyi–Kitaev operator formulas to molecular hydrogen in a four-spin-orbital minimal basis and prepares its electronic Hamiltonian for Pauli-matrix simulation.

  • VII. THE MOLECULAR ELECTRONIC HAMILTONIAN FOR THE HYDROGEN MOLECULE IN THE BRAVYI-KITAEV BASIS: The molecular electronic Hamiltonian is divided into one-electron and two-electron terms before conversion into Pauli products.The paper proceeds to simulate the resulting Hamiltonian after deriving the number-operator expressions.
  • VII. THE MOLECULAR ELECTRONIC HAMILTONIAN FOR THE HYDROGEN MOLECULE IN THE BRAVYI-KITAEV BASIS: Molecular hydrogen is treated in a minimal basis with four spin orbitals indexed 0 through 3.The Hamiltonian sums therefore run over these four spin orbitals.
  • VII. THE MOLECULAR ELECTRONIC HAMILTONIAN FOR THE HYDROGEN MOLECULE IN THE BRAVYI-KITAEV BASIS: The paper derives simplified Bravyi–Kitaev expressions for the individual terms of the molecular electronic Hamiltonian.The one- and two-electron Hamiltonians are considered separately.
  • VII. THE MOLECULAR ELECTRONIC HAMILTONIAN FOR THE HYDROGEN MOLECULE IN THE BRAVYI-KITAEV BASIS: The one- and two-electron overlap integrals are supplied for the four-orbital hydrogen calculation.The integrals were obtained using a restricted Hartree–Fock calculation in the PyQuante quantum chemistry package.
  • VII. THE MOLECULAR ELECTRONIC HAMILTONIAN FOR THE HYDROGEN MOLECULE IN THE BRAVYI-KITAEV BASIS: The hydrogen calculation uses an internuclear separation of 1.401000 atomic units.This corresponds to 7.414 × 10^-11 m.

B. The Bravyi-Kitaev Pauli representation of ˆH(2)

The paper converts the hydrogen electronic Hamiltonian into Bravyi–Kitaev Pauli products, derives its excitation terms, and compares the resulting simulation resources with Jordan–Wigner.

  • B. The Bravyi-Kitaev Pauli representation of ˆH(2): Coulomb and exchange contributions are expressed as products of number operators and reduced to Pauli products for the minimal-basis hydrogen Hamiltonian.The paper gives explicit examples for several indexed terms.
  • B. The Bravyi-Kitaev Pauli representation of ˆH(2): Double-excitation terms are derived by factoring them into products of operators of the form a†_i a_j and simplifying overlapping supports.The paper illustrates this procedure for the h0312 term and then repeats it for the second double excitation.
  • B. The Bravyi-Kitaev Pauli representation of ˆH(2): The H2 Hamiltonian is assembled from Pauli representations of number, Coulomb/exchange, and double-excitation operators.The construction uses the overlap integrals together with the operator expressions derived earlier.
  • B. The Bravyi-Kitaev Pauli representation of ˆH(2): The resulting Bravyi–Kitaev Hamiltonian is isospectral to the Jordan–Wigner-derived Hamiltonian.The two representations therefore describe the same spectrum while using different Pauli-product forms.
  • B. The Bravyi-Kitaev Pauli representation of ˆH(2): Writing both Hamiltonians as Pauli products enables comparison of the computational resources required for quantum simulation.Because not all Pauli tensor products commute, their exponentiation requires a Trotter approximation.

VIII. TROTTERIZATION

The paper decomposes Bravyi-Kitaev and Jordan-Wigner Hamiltonians into Pauli-product terms and approximates time evolution with Suzuki-Trotter formulas. For molecular hydrogen, Bravyi-Kitaev requires fewer gates to reach chemical precision, with savings depending on term ordering and Trotter order.

  • Trotterization: Suzuki-Trotter approximations replace the propagator by sequential exponentiation of Hamiltonian terms, with higher orders providing improved eigenvalue estimates as Trotter steps increase.The approximation becomes better as the number of Trotter steps n increases.
  • Hamiltonian structure: The Hamiltonian terms are divided into subsets whose terms mutually commute internally, while terms across subsets need not commute.This structure determines how the terms can be grouped during time evolution.
  • Gate accounting: Exponentiating an n-fold Pauli-Z product requires 2(n −1) CNOT gates plus one single-qubit gate.Parity is computed, a phase rotation is applied, and the parity is then uncomputed.
  • Gate accounting: Pauli-X and Pauli-Y factors require basis-changing single-qubit gates before parity computation and inverse gates during uncomputation.Each non-σz factor adds 2 single-qubit gates to exponentiation.
  • Ordering effects: The number of gates needed depends on term ordering, and optimizing ordering is difficult because electronic Hamiltonians contain O(n4) terms.The paper uses a naïve ordering for one comparison and coefficient-magnitude ordering as a more sophisticated variant.
  • Gate comparison: 74 gates per first-order Trotter step are required for Bravyi-Kitaev, compared with 82 for Jordan-Wigner.The comparison applies to either first-order variant because the per-step gate counts are unchanged.
  • Chemical precision: 222 gates achieve chemical precision for Bravyi-Kitaev, versus 328 gates for Jordan-Wigner when noncommuting terms are intermixed.The corresponding estimates use 3 and 4 Trotter steps, respectively.

IX. CONCLUSIONS

The paper develops a detailed Bravyi-Kitaev mapping for quantum chemical Hamiltonians and finds fewer gates than Jordan-Wigner for H2 in a minimal basis. Its asymptotically better scaling implies increasing advantages for larger basis sets and molecules.

  • The paper works out a detailed Bravyi-Kitaev transformation for Hermitian second-quantized operators in quantum chemical Hamiltonians.
  • 222 gates versus 328 gates are required for the H2 minimal-basis electron-dynamics simulation algorithm.
  • For minimal-basis H2, both derived spin Hamiltonians contain four-local Pauli tensor products, so the Bravyi-Kitaev simulation cost is only slightly reduced.
  • With eight molecular orbitals, Jordan-Wigner can produce up to eight-local Pauli products, whereas the Bravyi-Kitaev spin Hamiltonian would not.
  • O(log n) scaling for Bravyi-Kitaev versus O(n) for Jordan-Wigner makes their efficiency difference larger for larger basis sets and molecules.
  • The authors conclude that Bravyi-Kitaev simulations can make larger molecules and larger basis sets more accessible to experiment.
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