Quantum codes for quantum simulation of fermions on a square lattice of qubits

Steudtner, Mark; Wehner, Stephanie

doi:10.1103/physreva.99.022308

Cited by 55 publications

(45 citation statements)

References 56 publications

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Section: Proofmentioning

confidence: 73%

“…Since each of those gates is controlled by a different qubit, the only thing hindering a massive parallelization of V is the structure of the p strings themselves: the complexity of (5) depends on their individual Pauli weight and whether they overlap with one another. Fortunately, there is a considerable body of theoretical work on how to produce local strings when simulating fermions [17][18][19][20][21][22] and bosons [23]. While the decompressed data structure of the unary encoding benefits the complexity of V , benefits for the other components have not been shown: regardless of the encoding, S is suggested to be implemented by the sequence G • (2|0 0| − I) • G † [5,16], with the reflection (2|0 0| − I) requiring a Toffoli-type gate across the entire qubitization register; see Fig.…”

Section: (B)mentioning

confidence: 99%

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Estimating exact energies in quantum simulation without Toffoli gates

Steudtner

Wehner

2020

Phys. Rev. A

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Qubitization is a modern approach to estimate Hamiltonian eigenvalues without simulating its time evolution. While in this way approximation errors are avoided, its resource and gate requirements are more extensive: qubitization requires additional qubits to store information about the Hamiltonian, and Toffoli gates to probe them throughout the routine. Recently, it was shown that storing the Hamiltonian in a unary representation can alleviate the need for such gates in one of the qubitization subroutines. Building on that principle, we develop an alternative decomposition of the entire algorithm: without Toffoli gates, we can encode the Hamiltonian into qubits within logarithmic depth.

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Section: Proofmentioning

confidence: 73%

Section: (B)mentioning

confidence: 99%

Estimating exact energies in quantum simulation without Toffoli gates

Steudtner

Wehner

2020

Phys. Rev. A

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“…One strategy, employed by encodings presented in Refs. [10][11][12], leverages the Jordan-Wigner transform, and defines stabilizers that "cancel out" sections of the long strings, discarding a portion of the fermionic modes in the process.…”

Section: The Compact Encodingmentioning

confidence: 99%

“…There currently exist a handful of local fermionic encodings [10][11][12][13][14][15]. A comparison of these encodings is given in Table I.…”

Section: Introductionmentioning

confidence: 99%

Compact fermion to qubit mappings

et al. 2021

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Mappings between fermions and qubits are valuable constructions in physics. To date only a handful exist. In addition to revealing dualities between fermionic and spin systems, such mappings are indispensable in any quantum simulation of fermionic physics on quantum computers. The number of qubits required per fermionic mode, and the locality of mapped fermionic operators strongly impact the cost of such simulations. We present a fermion to qubit mapping that outperforms all previous local mappings in both the qubit to mode ratio and the locality of mapped operators. In addition to these practically useful features, the mapping bears an elegant relationship to the toric code, which we discuss. Finally, we consider the error mitigating properties of the mapping-which encodes fermionic states into the code space of a stabilizer code. Although there is an implicit tradeoff between low weight representations of local fermionic operators, and high distance code spaces, we argue that fermionic encodings with low-weight representations of local fermionic operators can still exhibit error mitigating properties which can serve a similar role to that played by high code distances. In particular, when undetectable errors correspond to "natural" fermionic noise. We illustrate this point explicitly both for this encoding and the Verstraete-Cirac encoding.

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“…Ball [34] [36]. Recently, Steudtner and Wehner [37] constructed a class of localitypreserving fermion-to-qubit mappings by concatenating the JW transformation with quantum codes. Some applications of locality-preserving mappings to quantum simulation were discussed in [30,38].…”

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confidence: 99%

Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations

et al. 2019

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Locality-preserving fermion-to-qubit mappings are especially useful for simulating lattice fermion models (e.g., the Hubbard model) on a quantum computer. They avoid the overhead associated with non-local parity terms in mappings such as the Jordan-Wigner transformation. As a result, they often provide solutions with lower circuit depth and gate complexity. A major obstacle to achieving near-term quantum computation is quantum noises. Interestingly, locality-preserving mappings encode the fermionic state in the common +1 eigenstate of a set of stabilizers, akin to quantum error-correcting codes. Here, we discuss a couple of known locality-preserving mappings and their abilities to correct/detect single-qubit errors. We also introduce a locality-preserving map, whose stabilizers are products of Majorana operators on closed paths of the fermionic hopping graph. The code can correct all single-qubit errors on a 2-dimensional square lattice, while previous localitypreserving codes can only detect single-qubit errors on the same lattice. Our code also has the advantage of having lower-weight logical operators. We expect that error-mitigating schemes with low overhead to be useful to the success of near-term quantum algorithms such as the variational quantum eigensolver.We are closer to realizing the potential of quantum computation [1-3] with the recent rapid advances in quantum computing devices such as ion traps [4-6] and superconducting qubits [7,8]. While performing faulttolerant quantum computation [9] is widely regarded as the ultimate goal, the substantial overheads prevent it from being implemented in the immediate future [10,11]. In the meantime, error-mitigation schemes are likely to be an essential component for the success of near-term quantum algorithms [12][13][14][15][16][17][18][19].Fermionic systems must be mapped to spin systems before they can be simulated on a digital quantum computer [20]. Mappings such as the Jordan-Wigner (JW) transformation introduce nonlocal parity terms when the spatial dimension is greater than one. These terms add considerable overhead to quantum simulations of local fermionic systems. Clever circuit compilation methods have been introduced to reduce the overhead of the parity terms on digital quantum computers with all-to-all connections [21,22].The non-locality resulting from parity becomes more prominent on near-term quantum devices, where only geometrically local two-qubit gates are available [23,24]. To overcome this difficulty, the fermionic SWAP gate can be used to bring together fermonic modes encoded far apart in the JW transformation [25]. It was shown that fermionic SWAP network offers asymptotically optimal solution to simulating quantum chemistry problems in terms of circuit depth [26][27][28]. For other problems, such as the two-dimensional (2D) fermionic Fourier transformation, the fermionic SWAP network is not optimal; this transformation can be implemented with a quadratically shorter circuit depth by going between different bases [29]. * qzj@google.com Loc...

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Quantum codes for quantum simulation of fermions on a square lattice of qubits

Cited by 55 publications

References 56 publications

Estimating exact energies in quantum simulation without Toffoli gates

Estimating exact energies in quantum simulation without Toffoli gates

Compact fermion to qubit mappings

Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations

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