Minimum Trotterization Formulas for a Time-Dependent Hamiltonian
When a time propagator $e^{\delta t A}$ for duration $\delta t$ consists of two noncommuting parts $A=X+Y$, Trotterization approximately decomposes the propagator into a product of exponentials of $X$ and $Y$. Various Trotterization formulas have been utilized in quantum and classical computers, but...
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| Format: | Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2023-11-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2023-11-06-1168/pdf/ |
| _version_ | 1797635687208452096 |
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| author | Tatsuhiko N. Ikeda Asir Abrar Isaac L. Chuang Sho Sugiura |
| author_facet | Tatsuhiko N. Ikeda Asir Abrar Isaac L. Chuang Sho Sugiura |
| author_sort | Tatsuhiko N. Ikeda |
| collection | DOAJ |
| description | When a time propagator $e^{\delta t A}$ for duration $\delta t$ consists of two noncommuting parts $A=X+Y$, Trotterization approximately decomposes the propagator into a product of exponentials of $X$ and $Y$. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator $A(t)$. Here, for $A(t)$ given by the sum of two operators $X$ and $Y$ with time-dependent coefficients $A(t) = x(t) X + y(t) Y$, we develop a systematic approach to derive high-order Trotterization formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we numerically benchmark the fourth-order formulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9-exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula. |
| first_indexed | 2024-03-11T12:23:57Z |
| format | Article |
| id | doaj.art-9413bb42f59c4e06a472678b7ec3b8ec |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-03-11T12:23:57Z |
| publishDate | 2023-11-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-9413bb42f59c4e06a472678b7ec3b8ec2023-11-06T13:54:24ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2023-11-017116810.22331/q-2023-11-06-116810.22331/q-2023-11-06-1168Minimum Trotterization Formulas for a Time-Dependent HamiltonianTatsuhiko N. IkedaAsir AbrarIsaac L. ChuangSho SugiuraWhen a time propagator $e^{\delta t A}$ for duration $\delta t$ consists of two noncommuting parts $A=X+Y$, Trotterization approximately decomposes the propagator into a product of exponentials of $X$ and $Y$. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator $A(t)$. Here, for $A(t)$ given by the sum of two operators $X$ and $Y$ with time-dependent coefficients $A(t) = x(t) X + y(t) Y$, we develop a systematic approach to derive high-order Trotterization formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we numerically benchmark the fourth-order formulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9-exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula.https://quantum-journal.org/papers/q-2023-11-06-1168/pdf/ |
| spellingShingle | Tatsuhiko N. Ikeda Asir Abrar Isaac L. Chuang Sho Sugiura Minimum Trotterization Formulas for a Time-Dependent Hamiltonian Quantum |
| title | Minimum Trotterization Formulas for a Time-Dependent Hamiltonian |
| title_full | Minimum Trotterization Formulas for a Time-Dependent Hamiltonian |
| title_fullStr | Minimum Trotterization Formulas for a Time-Dependent Hamiltonian |
| title_full_unstemmed | Minimum Trotterization Formulas for a Time-Dependent Hamiltonian |
| title_short | Minimum Trotterization Formulas for a Time-Dependent Hamiltonian |
| title_sort | minimum trotterization formulas for a time dependent hamiltonian |
| url | https://quantum-journal.org/papers/q-2023-11-06-1168/pdf/ |
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