Explore our research publications
Explore our research publications
Explore our research publications
Pioneering
Quantum Insights
Pioneering
Quantum Insights
Discover a curated selection of publications showcasing our breakthroughs in quantum research and computational science. Stay updated with fresh insights and innovations from our expert team.
Discover a curated selection of publications showcasing our breakthroughs in quantum research and computational science. Stay updated with fresh insights and innovations from our expert team.
Discover a curated selection of publications showcasing our breakthroughs in quantum research and computational science. Stay updated with fresh insights and innovations from our expert team.
Higher order tensor factorizations for block encoding vibrational and vibronic Hamiltonians
Hirsh Kamakari, Emil Żak
Fault tolerant quantum simulation via the phase estimation algorithm and qubitization has a T-gate count that scales proportionally to the 1-norm of the Hamiltonian, the cost of block encoding the Hamiltonian, and inversely proportionally to the desired accuracy. Tensor factorization methods have been successfully used to reduce T-gate counts in the ground state electronic structure problem. Here we introduce the use of tensor factorization methods to reduce the T-gate count of quantum phase estimation. In particular, we show how Canonical Polyadic and Tucker decompositions of the tensors representing the vibrational and vibronic Hamiltonians can be utilized to rewrite the Hamiltonian in terms of linear combination of bosonic position operators representing nuclear vibrations. We demonstrate the use of these factorization methods on the water and monodeutered methane molecules.
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Higher order tensor factorizations for block encoding vibrational and vibronic Hamiltonians
Hirsh Kamakari, Emil Żak
Fault tolerant quantum simulation via the phase estimation algorithm and qubitization has a T-gate count that scales proportionally to the 1-norm of the Hamiltonian, the cost of block encoding the Hamiltonian, and inversely proportionally to the desired accuracy. Tensor factorization methods have been successfully used to reduce T-gate counts in the ground state electronic structure problem. Here we introduce the use of tensor factorization methods to reduce the T-gate count of quantum phase estimation. In particular, we show how Canonical Polyadic and Tucker decompositions of the tensors representing the vibrational and vibronic Hamiltonians can be utilized to rewrite the Hamiltonian in terms of linear combination of bosonic position operators representing nuclear vibrations. We demonstrate the use of these factorization methods on the water and monodeutered methane molecules.
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Higher order tensor factorizations for block encoding vibrational and vibronic Hamiltonians
Hirsh Kamakari, Emil Żak
Fault tolerant quantum simulation via the phase estimation algorithm and qubitization has a T-gate count that scales proportionally to the 1-norm of the Hamiltonian, the cost of block encoding the Hamiltonian, and inversely proportionally to the desired accuracy. Tensor factorization methods have been successfully used to reduce T-gate counts in the ground state electronic structure problem. Here we introduce the use of tensor factorization methods to reduce the T-gate count of quantum phase estimation. In particular, we show how Canonical Polyadic and Tucker decompositions of the tensors representing the vibrational and vibronic Hamiltonians can be utilized to rewrite the Hamiltonian in terms of linear combination of bosonic position operators representing nuclear vibrations. We demonstrate the use of these factorization methods on the water and monodeutered methane molecules.
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Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
Berta Casas, Grzegorz Rajchel-Mieldzioć, Suhail Ahmad Rather, Marcin Płodzień, Wojciech Bruzda, Alba Cervera-Lierta, Karol Życzkowski
Absolutely maximally entangled (AME) states of multipartite quantum systems exhibit maximal entanglement across all possible bipartitions. These states lead to teleportation protocols that surpass standard teleportation schemes, determine quantum error correction codes and can be used to test performance of current term quantum processors. Several AME states can be constructed from graph states using minimal quantum resources. However, there exist other constructions that depart from the stabilizer formalism. In this work, we present explicit quantum circuits to generate exemplary non-stabilizer AME states of four subsystems with four, six, and eight levels each and analyze their capabilities to perform quantum information tasks.
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Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
Berta Casas, Grzegorz Rajchel-Mieldzioć, Suhail Ahmad Rather, Marcin Płodzień, Wojciech Bruzda, Alba Cervera-Lierta, Karol Życzkowski
Absolutely maximally entangled (AME) states of multipartite quantum systems exhibit maximal entanglement across all possible bipartitions. These states lead to teleportation protocols that surpass standard teleportation schemes, determine quantum error correction codes and can be used to test performance of current term quantum processors. Several AME states can be constructed from graph states using minimal quantum resources. However, there exist other constructions that depart from the stabilizer formalism. In this work, we present explicit quantum circuits to generate exemplary non-stabilizer AME states of four subsystems with four, six, and eight levels each and analyze their capabilities to perform quantum information tasks.
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Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
Berta Casas, Grzegorz Rajchel-Mieldzioć, Suhail Ahmad Rather, Marcin Płodzień, Wojciech Bruzda, Alba Cervera-Lierta, Karol Życzkowski
Absolutely maximally entangled (AME) states of multipartite quantum systems exhibit maximal entanglement across all possible bipartitions. These states lead to teleportation protocols that surpass standard teleportation schemes, determine quantum error correction codes and can be used to test performance of current term quantum processors. Several AME states can be constructed from graph states using minimal quantum resources. However, there exist other constructions that depart from the stabilizer formalism. In this work, we present explicit quantum circuits to generate exemplary non-stabilizer AME states of four subsystems with four, six, and eight levels each and analyze their capabilities to perform quantum information tasks.
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Classical Simulation of Quantum CSP Strategies
Demian Banakh, Lorenzo Ciardo, Marcin Kozik, Jan Tułowiecki
We prove that any perfect quantum strategy for the two-prover game encoding a constraint satisfaction problem (CSP) can be simulated via a perfect classical strategy with an extra classical communication channel, whose size depends only on (i) the size of the shared quantum system used in the quantum strategy, and (ii) structural parameters of the CSP template. The result is obtained via a combinatorial characterisation of perfect classical strategies with extra communication channels and a geometric rounding procedure for the projection-valued measurements involved in quantum strategies. A key intermediate step of our proof is to establish that the gap between the classical chromatic number of graphs and its quantum variant is bounded when the quantum strategy involves shared quantum information of bounded size.
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Classical Simulation of Quantum CSP Strategies
Demian Banakh, Lorenzo Ciardo, Marcin Kozik, Jan Tułowiecki
We prove that any perfect quantum strategy for the two-prover game encoding a constraint satisfaction problem (CSP) can be simulated via a perfect classical strategy with an extra classical communication channel, whose size depends only on (i) the size of the shared quantum system used in the quantum strategy, and (ii) structural parameters of the CSP template. The result is obtained via a combinatorial characterisation of perfect classical strategies with extra communication channels and a geometric rounding procedure for the projection-valued measurements involved in quantum strategies. A key intermediate step of our proof is to establish that the gap between the classical chromatic number of graphs and its quantum variant is bounded when the quantum strategy involves shared quantum information of bounded size.
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Classical Simulation of Quantum CSP Strategies
Demian Banakh, Lorenzo Ciardo, Marcin Kozik, Jan Tułowiecki
We prove that any perfect quantum strategy for the two-prover game encoding a constraint satisfaction problem (CSP) can be simulated via a perfect classical strategy with an extra classical communication channel, whose size depends only on (i) the size of the shared quantum system used in the quantum strategy, and (ii) structural parameters of the CSP template. The result is obtained via a combinatorial characterisation of perfect classical strategies with extra communication channels and a geometric rounding procedure for the projection-valued measurements involved in quantum strategies. A key intermediate step of our proof is to establish that the gap between the classical chromatic number of graphs and its quantum variant is bounded when the quantum strategy involves shared quantum information of bounded size.
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Simultaneously optimizing symmetry shifts and tensor factorizations for cost-efficient Fault-Tolerant Quantum Simulations of electronic Hamiltonians
Konrad Deka, Emil Zak
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor-factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous techniques. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the feasiblity of practical Hamiltonian simulations. We demonstrate the effectiveness of our technique on Hamiltonians of industrial and biological relevance, including the nitrogenase cofactor (FeMoCo) and cytochrome P450.
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Simultaneously optimizing symmetry shifts and tensor factorizations for cost-efficient Fault-Tolerant Quantum Simulations of electronic Hamiltonians
Konrad Deka, Emil Zak
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor-factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous techniques. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the feasiblity of practical Hamiltonian simulations. We demonstrate the effectiveness of our technique on Hamiltonians of industrial and biological relevance, including the nitrogenase cofactor (FeMoCo) and cytochrome P450.
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Simultaneously optimizing symmetry shifts and tensor factorizations for cost-efficient Fault-Tolerant Quantum Simulations of electronic Hamiltonians
Konrad Deka, Emil Zak
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor-factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous techniques. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the feasiblity of practical Hamiltonian simulations. We demonstrate the effectiveness of our technique on Hamiltonians of industrial and biological relevance, including the nitrogenase cofactor (FeMoCo) and cytochrome P450.
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Quantum Error Detection For Early Term Fault-Tolerant Quantum Algorithms
Tom Ginsberg, Vyom Patel
Quantum error detection (QED) offers a promising pathway to fault tolerance in near-term quantum devices by balancing error suppression with minimal resource overhead. However, its practical utility hinges on optimizing design parameters-such as syndrome measurement frequency-to avoid diminishing returns from detection overhead. In this work, we present a comprehensive framework for fault-tolerant compilation and simulation of quantum algorithms using [[n, n-2, 2]] codes, which enable low-qubit-overhead error detection and a simple nearly fault-tolerant universal set of operations. We demonstrate and analyze our pipeline with a purely statistical interpretation and through the implementation of Grover's search algorithm. Our results are used to answer the question is quantum error detection a worthwhile avenue for early-term fault tolerance, and if so how can we get the most out of it? Simulations under the circuit-level noise model reveal that finding optimal syndrome schedules improves algorithm success probabilities by an average of 6.7x but eventual statistical limits from post-selection in noisy/resource-limited regimes constrain scalability. Furthermore, we propose a simple data-driven approach to predict fault tolerant compilation parameters, such as optimal syndrome schedules, and expected fault tolerant performance gains based on circuit and noise features. These results provide actionable guidelines for implementing QED in early-term quantum experiments and underscore its role as a pragmatic, constant-overhead error mitigation layer for shallow algorithms. To aid in further research, we release all simulation data computed for this work and provide an experimental QED compiler at https://codeqraft.xyz/qed.
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Quantum Error Detection For Early Term Fault-Tolerant Quantum Algorithms
Tom Ginsberg, Vyom Patel
Quantum error detection (QED) offers a promising pathway to fault tolerance in near-term quantum devices by balancing error suppression with minimal resource overhead. However, its practical utility hinges on optimizing design parameters-such as syndrome measurement frequency-to avoid diminishing returns from detection overhead. In this work, we present a comprehensive framework for fault-tolerant compilation and simulation of quantum algorithms using [[n, n-2, 2]] codes, which enable low-qubit-overhead error detection and a simple nearly fault-tolerant universal set of operations. We demonstrate and analyze our pipeline with a purely statistical interpretation and through the implementation of Grover's search algorithm. Our results are used to answer the question is quantum error detection a worthwhile avenue for early-term fault tolerance, and if so how can we get the most out of it? Simulations under the circuit-level noise model reveal that finding optimal syndrome schedules improves algorithm success probabilities by an average of 6.7x but eventual statistical limits from post-selection in noisy/resource-limited regimes constrain scalability. Furthermore, we propose a simple data-driven approach to predict fault tolerant compilation parameters, such as optimal syndrome schedules, and expected fault tolerant performance gains based on circuit and noise features. These results provide actionable guidelines for implementing QED in early-term quantum experiments and underscore its role as a pragmatic, constant-overhead error mitigation layer for shallow algorithms. To aid in further research, we release all simulation data computed for this work and provide an experimental QED compiler at https://codeqraft.xyz/qed.
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Quantum Error Detection For Early Term Fault-Tolerant Quantum Algorithms
Tom Ginsberg, Vyom Patel
Quantum error detection (QED) offers a promising pathway to fault tolerance in near-term quantum devices by balancing error suppression with minimal resource overhead. However, its practical utility hinges on optimizing design parameters-such as syndrome measurement frequency-to avoid diminishing returns from detection overhead. In this work, we present a comprehensive framework for fault-tolerant compilation and simulation of quantum algorithms using [[n, n-2, 2]] codes, which enable low-qubit-overhead error detection and a simple nearly fault-tolerant universal set of operations. We demonstrate and analyze our pipeline with a purely statistical interpretation and through the implementation of Grover's search algorithm. Our results are used to answer the question is quantum error detection a worthwhile avenue for early-term fault tolerance, and if so how can we get the most out of it? Simulations under the circuit-level noise model reveal that finding optimal syndrome schedules improves algorithm success probabilities by an average of 6.7x but eventual statistical limits from post-selection in noisy/resource-limited regimes constrain scalability. Furthermore, we propose a simple data-driven approach to predict fault tolerant compilation parameters, such as optimal syndrome schedules, and expected fault tolerant performance gains based on circuit and noise features. These results provide actionable guidelines for implementing QED in early-term quantum experiments and underscore its role as a pragmatic, constant-overhead error mitigation layer for shallow algorithms. To aid in further research, we release all simulation data computed for this work and provide an experimental QED compiler at https://codeqraft.xyz/qed.
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Utilizing redundancies in Qubit Hilbert Space to reduce entangling gate counts in the Unitary Vibrational Coupled-Cluster Method
Michał Szczepanik, Emil Żak
We present a new method for state preparation using the Unitary Vibrational Coupled-Cluster (UVCC) technique. Our approach utilizes redundancies in the Hilbert space in the direct mapping of vibrational modes into qubits. By eliminating half of the qubit controls required in the Trotterized UVCC ansatz, our method achieves up to a 50% theoretical reduction in the entangling gate count compared to other methods and up to a 28% reduction compared practically useful approaches. This improvement enhances the fidelity of UVCC state preparation, enabling more efficient and earlier implementation of complex quantum vibrational structure calculations on near-term quantum devices. We experimentally demonstrate our method on Quantinuum's H1-1 quantum hardware, achieving significantly higher fidelities for 6- and 8-qubit systems compared to existing implementations. For fault-tolerant architectures, eliminating half of the control qubits in multi-controlled rotations incurs an additional Toffoli gate overhead elsewhere in the circuit. Thus, the overall performance gain depends on the specific decomposition method used for multi-controlled gates.
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Utilizing redundancies in Qubit Hilbert Space to reduce entangling gate counts in the Unitary Vibrational Coupled-Cluster Method
Michał Szczepanik, Emil Żak
We present a new method for state preparation using the Unitary Vibrational Coupled-Cluster (UVCC) technique. Our approach utilizes redundancies in the Hilbert space in the direct mapping of vibrational modes into qubits. By eliminating half of the qubit controls required in the Trotterized UVCC ansatz, our method achieves up to a 50% theoretical reduction in the entangling gate count compared to other methods and up to a 28% reduction compared practically useful approaches. This improvement enhances the fidelity of UVCC state preparation, enabling more efficient and earlier implementation of complex quantum vibrational structure calculations on near-term quantum devices. We experimentally demonstrate our method on Quantinuum's H1-1 quantum hardware, achieving significantly higher fidelities for 6- and 8-qubit systems compared to existing implementations. For fault-tolerant architectures, eliminating half of the control qubits in multi-controlled rotations incurs an additional Toffoli gate overhead elsewhere in the circuit. Thus, the overall performance gain depends on the specific decomposition method used for multi-controlled gates.
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Utilizing redundancies in Qubit Hilbert Space to reduce entangling gate counts in the Unitary Vibrational Coupled-Cluster Method
Michał Szczepanik, Emil Żak
We present a new method for state preparation using the Unitary Vibrational Coupled-Cluster (UVCC) technique. Our approach utilizes redundancies in the Hilbert space in the direct mapping of vibrational modes into qubits. By eliminating half of the qubit controls required in the Trotterized UVCC ansatz, our method achieves up to a 50% theoretical reduction in the entangling gate count compared to other methods and up to a 28% reduction compared practically useful approaches. This improvement enhances the fidelity of UVCC state preparation, enabling more efficient and earlier implementation of complex quantum vibrational structure calculations on near-term quantum devices. We experimentally demonstrate our method on Quantinuum's H1-1 quantum hardware, achieving significantly higher fidelities for 6- and 8-qubit systems compared to existing implementations. For fault-tolerant architectures, eliminating half of the control qubits in multi-controlled rotations incurs an additional Toffoli gate overhead elsewhere in the circuit. Thus, the overall performance gain depends on the specific decomposition method used for multi-controlled gates.
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Simultaneously Optimizing Symmetry Shifts and Tensor Factorizations for Cost-Efficient Fault-Tolerant Quantum Simulations of Electronic Hamiltonians
Konrad Deka, Emil Żak
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor-factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous techniques. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the feasiblity of practical Hamiltonian simulations. We demonstrate the effectiveness of our technique on Hamiltonians of industrial and biological relevance, including the nitrogenase cofactor (FeMoCo) and cytochrome P450.
Download
Simultaneously Optimizing Symmetry Shifts and Tensor Factorizations for Cost-Efficient Fault-Tolerant Quantum Simulations of Electronic Hamiltonians
Konrad Deka, Emil Żak
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor-factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous techniques. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the feasiblity of practical Hamiltonian simulations. We demonstrate the effectiveness of our technique on Hamiltonians of industrial and biological relevance, including the nitrogenase cofactor (FeMoCo) and cytochrome P450.
Download
Simultaneously Optimizing Symmetry Shifts and Tensor Factorizations for Cost-Efficient Fault-Tolerant Quantum Simulations of Electronic Hamiltonians
Konrad Deka, Emil Żak
In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor-factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous techniques. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the feasiblity of practical Hamiltonian simulations. We demonstrate the effectiveness of our technique on Hamiltonians of industrial and biological relevance, including the nitrogenase cofactor (FeMoCo) and cytochrome P450.
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Spoofing of Quantum Channels Enables Low-Rank Projective Simulation
Timothy Heightman, Grzegorz Rajchel-Mieldzioć
The ability to characterise and discern quantum channels is a crucial aspect of noisy quantum technologies. In this work, we explore the problem of distinguishing quantum channels when limited to sub-exponential resources, framed as von Neumann (projective) measurements. We completely characterise equivalence classes of quantum channels with different Kraus ranks that have the same marginal distributions under compatible projective measurements. In doing so, we explicitly identify gauge freedoms which can be varied without changing those compatible marginal outcome distributions, opening new avenues for quantum channel simulation, variational quantum channels, as well as novel adversarial strategies in noisy quantum device certification. Specifically, we show how a Sinkhorn-like algorithm enables us to find the minimum admissible Kraus rank that generates the correct outcome marginals. For a generic d-dimensional quantum system, this lowers the Kraus rank from d^2 to the theoretical minimum of d. For up to d=20, we numerically demonstrate our findings, for which the code is available and open source. Finally, we provide an analytic algorithm for the special case of spoofing Pauli channels.
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Spoofing of Quantum Channels Enables Low-Rank Projective Simulation
Timothy Heightman, Grzegorz Rajchel-Mieldzioć
The ability to characterise and discern quantum channels is a crucial aspect of noisy quantum technologies. In this work, we explore the problem of distinguishing quantum channels when limited to sub-exponential resources, framed as von Neumann (projective) measurements. We completely characterise equivalence classes of quantum channels with different Kraus ranks that have the same marginal distributions under compatible projective measurements. In doing so, we explicitly identify gauge freedoms which can be varied without changing those compatible marginal outcome distributions, opening new avenues for quantum channel simulation, variational quantum channels, as well as novel adversarial strategies in noisy quantum device certification. Specifically, we show how a Sinkhorn-like algorithm enables us to find the minimum admissible Kraus rank that generates the correct outcome marginals. For a generic d-dimensional quantum system, this lowers the Kraus rank from d^2 to the theoretical minimum of d. For up to d=20, we numerically demonstrate our findings, for which the code is available and open source. Finally, we provide an analytic algorithm for the special case of spoofing Pauli channels.
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Spoofing of Quantum Channels Enables Low-Rank Projective Simulation
Timothy Heightman, Grzegorz Rajchel-Mieldzioć
The ability to characterise and discern quantum channels is a crucial aspect of noisy quantum technologies. In this work, we explore the problem of distinguishing quantum channels when limited to sub-exponential resources, framed as von Neumann (projective) measurements. We completely characterise equivalence classes of quantum channels with different Kraus ranks that have the same marginal distributions under compatible projective measurements. In doing so, we explicitly identify gauge freedoms which can be varied without changing those compatible marginal outcome distributions, opening new avenues for quantum channel simulation, variational quantum channels, as well as novel adversarial strategies in noisy quantum device certification. Specifically, we show how a Sinkhorn-like algorithm enables us to find the minimum admissible Kraus rank that generates the correct outcome marginals. For a generic d-dimensional quantum system, this lowers the Kraus rank from d^2 to the theoretical minimum of d. For up to d=20, we numerically demonstrate our findings, for which the code is available and open source. Finally, we provide an analytic algorithm for the special case of spoofing Pauli channels.
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Novel oracle constructions for quantum random access memory
Ákos Nagy, Cindy Zhang
A New Approach to Quantum Random Access Memory (QRAM) Discover our innovative design for QRAM that leverages the Walsh-Hadamard Transform to optimize circuit efficiency. Our method adapts the complexity of the circuit based on the transform's sparsity rather than the function itself—ideal for binary optimization and other low-degree functions. With a flexible design that lets you trade circuit depth for qubit count, our QRAM circuits achieve significant performance improvements, even for challenging Boolean functions. Download the full paper to learn more about these scalable and efficient QRAM constructions.
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Novel oracle constructions for quantum random access memory
Ákos Nagy, Cindy Zhang
A New Approach to Quantum Random Access Memory (QRAM) Discover our innovative design for QRAM that leverages the Walsh-Hadamard Transform to optimize circuit efficiency. Our method adapts the complexity of the circuit based on the transform's sparsity rather than the function itself—ideal for binary optimization and other low-degree functions. With a flexible design that lets you trade circuit depth for qubit count, our QRAM circuits achieve significant performance improvements, even for challenging Boolean functions. Download the full paper to learn more about these scalable and efficient QRAM constructions.
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Novel oracle constructions for quantum random access memory
Ákos Nagy, Cindy Zhang
A New Approach to Quantum Random Access Memory (QRAM) Discover our innovative design for QRAM that leverages the Walsh-Hadamard Transform to optimize circuit efficiency. Our method adapts the complexity of the circuit based on the transform's sparsity rather than the function itself—ideal for binary optimization and other low-degree functions. With a flexible design that lets you trade circuit depth for qubit count, our QRAM circuits achieve significant performance improvements, even for challenging Boolean functions. Download the full paper to learn more about these scalable and efficient QRAM constructions.
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Diagonal operator decomposition on restricted topologies via enumeration of quantum state subsets
Jan Tułowiecki, Łukasz Czerwiński, Konrad Deka, Jan Gwinner, Witold Jarnicki, Adam Szady
Various quantum algorithms require usage of arbitrary diagonal operators as subroutines. For their execution on a physical hardware, those operators must be first decomposed into target device's native gateset and its qubit connectivity for entangling gates. Here, we assume that the allowed gates are exactly the CX gate and the parameterized phase gate. We introduce a framework for the analysis of CX-only circuits and through its lens provide solution constructions for several different device topologies (fully-connected, linear and circular). We also introduce two additional variants of the problem. Those variants can be used in place of exact decomposition of the diagonal operator when the circuit following it satisfies a set of prerequisites, enabling further reduction in the CX cost of implementation. Finally, we discuss how to exploit the framework for the decomposition of a particular, rather than general, diagonal operator.
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Diagonal operator decomposition on restricted topologies via enumeration of quantum state subsets
Jan Tułowiecki, Łukasz Czerwiński, Konrad Deka, Jan Gwinner, Witold Jarnicki, Adam Szady
Various quantum algorithms require usage of arbitrary diagonal operators as subroutines. For their execution on a physical hardware, those operators must be first decomposed into target device's native gateset and its qubit connectivity for entangling gates. Here, we assume that the allowed gates are exactly the CX gate and the parameterized phase gate. We introduce a framework for the analysis of CX-only circuits and through its lens provide solution constructions for several different device topologies (fully-connected, linear and circular). We also introduce two additional variants of the problem. Those variants can be used in place of exact decomposition of the diagonal operator when the circuit following it satisfies a set of prerequisites, enabling further reduction in the CX cost of implementation. Finally, we discuss how to exploit the framework for the decomposition of a particular, rather than general, diagonal operator.
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Diagonal operator decomposition on restricted topologies via enumeration of quantum state subsets
Jan Tułowiecki, Łukasz Czerwiński, Konrad Deka, Jan Gwinner, Witold Jarnicki, Adam Szady
Various quantum algorithms require usage of arbitrary diagonal operators as subroutines. For their execution on a physical hardware, those operators must be first decomposed into target device's native gateset and its qubit connectivity for entangling gates. Here, we assume that the allowed gates are exactly the CX gate and the parameterized phase gate. We introduce a framework for the analysis of CX-only circuits and through its lens provide solution constructions for several different device topologies (fully-connected, linear and circular). We also introduce two additional variants of the problem. Those variants can be used in place of exact decomposition of the diagonal operator when the circuit following it satisfies a set of prerequisites, enabling further reduction in the CX cost of implementation. Finally, we discuss how to exploit the framework for the decomposition of a particular, rather than general, diagonal operator.
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Clifford circuits over non-cyclic abelian groups
Milo Moses, Jacek Horecki, Konrad Deka, Jan Tulowiecki
We present a discussion of the generalized Clifford group over non-cyclic finite abelian groups. These Clifford groups appear naturally in the theory of topological error correction and abelian anyon models. We demonstrate a generalized Gottesman-Knill theorem, stating that every Clifford circuit can be efficiently classically simulated. We additionally provide circuits for a universal quantum computing scheme based on local two-qudit Clifford gates and magic states.
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Clifford circuits over non-cyclic abelian groups
Milo Moses, Jacek Horecki, Konrad Deka, Jan Tulowiecki
We present a discussion of the generalized Clifford group over non-cyclic finite abelian groups. These Clifford groups appear naturally in the theory of topological error correction and abelian anyon models. We demonstrate a generalized Gottesman-Knill theorem, stating that every Clifford circuit can be efficiently classically simulated. We additionally provide circuits for a universal quantum computing scheme based on local two-qudit Clifford gates and magic states.
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Clifford circuits over non-cyclic abelian groups
Milo Moses, Jacek Horecki, Konrad Deka, Jan Tulowiecki
We present a discussion of the generalized Clifford group over non-cyclic finite abelian groups. These Clifford groups appear naturally in the theory of topological error correction and abelian anyon models. We demonstrate a generalized Gottesman-Knill theorem, stating that every Clifford circuit can be efficiently classically simulated. We additionally provide circuits for a universal quantum computing scheme based on local two-qudit Clifford gates and magic states.
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Fixed-point Grover Adaptive Search for Quadratic Binary Optimization Problems
Ákos Nagy, Jaime Park, Cindy Zhang, Atithi Acharya, Alex Khan
We explore a variant of Grover’s search tailored for solving Quadratic Unconstrained Binary Optimization (QUBO) problems. For a QUBO problem with n dimensions and m nonzero terms, we design a “marker” oracle that features a tunable parameter ranging from 1 to m. This oracle is built to work at a chosen precision level and uses a number of qubits that scales roughly with the problem size and the precision setting. Its overall circuit depth—especially the part requiring resource-intensive non-Clifford gates—is kept modest and, for all settings of the tunable parameter, uses at least about half as many non-Clifford gates as previous designs. In fact, for specific cases like maximum graph cut problems, a smart choice of precision allows us to reduce the circuit depth to grow only logarithmically with the problem size, while keeping qubit connectivity requirements minimal.
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Fixed-point Grover Adaptive Search for Quadratic Binary Optimization Problems
Ákos Nagy, Jaime Park, Cindy Zhang, Atithi Acharya, Alex Khan
We explore a variant of Grover’s search tailored for solving Quadratic Unconstrained Binary Optimization (QUBO) problems. For a QUBO problem with n dimensions and m nonzero terms, we design a “marker” oracle that features a tunable parameter ranging from 1 to m. This oracle is built to work at a chosen precision level and uses a number of qubits that scales roughly with the problem size and the precision setting. Its overall circuit depth—especially the part requiring resource-intensive non-Clifford gates—is kept modest and, for all settings of the tunable parameter, uses at least about half as many non-Clifford gates as previous designs. In fact, for specific cases like maximum graph cut problems, a smart choice of precision allows us to reduce the circuit depth to grow only logarithmically with the problem size, while keeping qubit connectivity requirements minimal.
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Fixed-point Grover Adaptive Search for Quadratic Binary Optimization Problems
Ákos Nagy, Jaime Park, Cindy Zhang, Atithi Acharya, Alex Khan
We explore a variant of Grover’s search tailored for solving Quadratic Unconstrained Binary Optimization (QUBO) problems. For a QUBO problem with n dimensions and m nonzero terms, we design a “marker” oracle that features a tunable parameter ranging from 1 to m. This oracle is built to work at a chosen precision level and uses a number of qubits that scales roughly with the problem size and the precision setting. Its overall circuit depth—especially the part requiring resource-intensive non-Clifford gates—is kept modest and, for all settings of the tunable parameter, uses at least about half as many non-Clifford gates as previous designs. In fact, for specific cases like maximum graph cut problems, a smart choice of precision allows us to reduce the circuit depth to grow only logarithmically with the problem size, while keeping qubit connectivity requirements minimal.
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Anyons in a highly-entangled toric xy model
Milo Moses, Konrad Deka
While ostensibly coined in 1989 by Xiao-Gang Wen, the term "topological order" has been in use since 1972 to describe the behavior of the classical xy model. It has been noted that the xy model does not have Wen's topological order since it is also subject a non-topological U(1) gauge action. We show in a sense this is the only obstruction. That is, if gauge invariance is enforced energetically then the xy model becomes purely topologically ordered. In fact, we show that the quantum xy topological order is an infinite lattice limit of Kitaev's quantum double model applied to the group G=Z.
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Anyons in a highly-entangled toric xy model
Milo Moses, Konrad Deka
While ostensibly coined in 1989 by Xiao-Gang Wen, the term "topological order" has been in use since 1972 to describe the behavior of the classical xy model. It has been noted that the xy model does not have Wen's topological order since it is also subject a non-topological U(1) gauge action. We show in a sense this is the only obstruction. That is, if gauge invariance is enforced energetically then the xy model becomes purely topologically ordered. In fact, we show that the quantum xy topological order is an infinite lattice limit of Kitaev's quantum double model applied to the group G=Z.
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Anyons in a highly-entangled toric xy model
Milo Moses, Konrad Deka
While ostensibly coined in 1989 by Xiao-Gang Wen, the term "topological order" has been in use since 1972 to describe the behavior of the classical xy model. It has been noted that the xy model does not have Wen's topological order since it is also subject a non-topological U(1) gauge action. We show in a sense this is the only obstruction. That is, if gauge invariance is enforced energetically then the xy model becomes purely topologically ordered. In fact, we show that the quantum xy topological order is an infinite lattice limit of Kitaev's quantum double model applied to the group G=Z.
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Creating rotational coherences in molecules aligned along the intermediate moment of inertia axis
Emil J. Zak
We propose and computationally study a method for simultaneously orienting the angular momentum of asymmetric top molecules along: 1) a laboratory-fixed direction; 2) the molecular intermediate moment of inertia axis; 3) the laser field wavevector. For this purpose we utilize a coherent control scheme in which a tailored-pulse optical centrifuge populates rotational states with well defined projections of the total angular momentum onto molecular axes. Appropriately time-shaped optical centrifuge pulses can leave the rotational wavepacket in peculiar rotational coherences which lead to a good degree of 3-dimensional transient alignment, with an arbitrary molecular axis pointing along the laser pulse propagation direction. As an example, we demonstrate how to generate highly resilient rotational quantum states of D2S in which the molecule rotates mainly about its intermediate inertia axis, such that its electric dipole moment is permanently aligned along the propagation direction of the laser pulse. Applications might include accessing less obscured information in various photo-electron imaging experiments.
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Creating rotational coherences in molecules aligned along the intermediate moment of inertia axis
Emil J. Zak
We propose and computationally study a method for simultaneously orienting the angular momentum of asymmetric top molecules along: 1) a laboratory-fixed direction; 2) the molecular intermediate moment of inertia axis; 3) the laser field wavevector. For this purpose we utilize a coherent control scheme in which a tailored-pulse optical centrifuge populates rotational states with well defined projections of the total angular momentum onto molecular axes. Appropriately time-shaped optical centrifuge pulses can leave the rotational wavepacket in peculiar rotational coherences which lead to a good degree of 3-dimensional transient alignment, with an arbitrary molecular axis pointing along the laser pulse propagation direction. As an example, we demonstrate how to generate highly resilient rotational quantum states of D2S in which the molecule rotates mainly about its intermediate inertia axis, such that its electric dipole moment is permanently aligned along the propagation direction of the laser pulse. Applications might include accessing less obscured information in various photo-electron imaging experiments.
Download
Creating rotational coherences in molecules aligned along the intermediate moment of inertia axis
Emil J. Zak
We propose and computationally study a method for simultaneously orienting the angular momentum of asymmetric top molecules along: 1) a laboratory-fixed direction; 2) the molecular intermediate moment of inertia axis; 3) the laser field wavevector. For this purpose we utilize a coherent control scheme in which a tailored-pulse optical centrifuge populates rotational states with well defined projections of the total angular momentum onto molecular axes. Appropriately time-shaped optical centrifuge pulses can leave the rotational wavepacket in peculiar rotational coherences which lead to a good degree of 3-dimensional transient alignment, with an arbitrary molecular axis pointing along the laser pulse propagation direction. As an example, we demonstrate how to generate highly resilient rotational quantum states of D2S in which the molecule rotates mainly about its intermediate inertia axis, such that its electric dipole moment is permanently aligned along the propagation direction of the laser pulse. Applications might include accessing less obscured information in various photo-electron imaging experiments.
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Efficient unstructured search implementation on current ion-trap quantum processors
Vladyslav Hlembotskyi, Rafał Burczyński, Witold Jarnicki, Adam Szady, Jan Tułowiecki
So far, only the results on 3 qubit spaces (both on superconducting and ion-trap realisations of quantum processors) have beaten the classical unstructured search in the expected number of oracle calls using optimal protocols in both settings. We present experimental results on running unstructured search in spaces defined by 4, 5 and 6 qubits on ion-trapped quantum processor. Our best circuits obtained respectively 66\%, 26\% and 6\% average probability of measuring the marked element. In the case of 4 and 5 qubit spaces we obtained fewer expected number of oracle calls required to find a marked element than any classical approach. Viability of the theoretical result by Grover at these qubit counts is, to authors' knowledge demonstrated experimentally for the first time. Also at 6 qubits, a circuit using a single oracle call returned a measured probability of success exceeding any possible classical approach. These results were achieved using a variety of unstructured search algorithms in conjunction with recent developments in reducing the number of entangling gates. The latter are currently considered to be a dominating source of errors in quantum computations. Some of these improvements have been made possible by using mid-circuit measurements. To our knowledge the latter feature is currently available only on the H0 quantum processor we run on.
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Efficient unstructured search implementation on current ion-trap quantum processors
Vladyslav Hlembotskyi, Rafał Burczyński, Witold Jarnicki, Adam Szady, Jan Tułowiecki
So far, only the results on 3 qubit spaces (both on superconducting and ion-trap realisations of quantum processors) have beaten the classical unstructured search in the expected number of oracle calls using optimal protocols in both settings. We present experimental results on running unstructured search in spaces defined by 4, 5 and 6 qubits on ion-trapped quantum processor. Our best circuits obtained respectively 66\%, 26\% and 6\% average probability of measuring the marked element. In the case of 4 and 5 qubit spaces we obtained fewer expected number of oracle calls required to find a marked element than any classical approach. Viability of the theoretical result by Grover at these qubit counts is, to authors' knowledge demonstrated experimentally for the first time. Also at 6 qubits, a circuit using a single oracle call returned a measured probability of success exceeding any possible classical approach. These results were achieved using a variety of unstructured search algorithms in conjunction with recent developments in reducing the number of entangling gates. The latter are currently considered to be a dominating source of errors in quantum computations. Some of these improvements have been made possible by using mid-circuit measurements. To our knowledge the latter feature is currently available only on the H0 quantum processor we run on.
Download
Efficient unstructured search implementation on current ion-trap quantum processors
Vladyslav Hlembotskyi, Rafał Burczyński, Witold Jarnicki, Adam Szady, Jan Tułowiecki
So far, only the results on 3 qubit spaces (both on superconducting and ion-trap realisations of quantum processors) have beaten the classical unstructured search in the expected number of oracle calls using optimal protocols in both settings. We present experimental results on running unstructured search in spaces defined by 4, 5 and 6 qubits on ion-trapped quantum processor. Our best circuits obtained respectively 66\%, 26\% and 6\% average probability of measuring the marked element. In the case of 4 and 5 qubit spaces we obtained fewer expected number of oracle calls required to find a marked element than any classical approach. Viability of the theoretical result by Grover at these qubit counts is, to authors' knowledge demonstrated experimentally for the first time. Also at 6 qubits, a circuit using a single oracle call returned a measured probability of success exceeding any possible classical approach. These results were achieved using a variety of unstructured search algorithms in conjunction with recent developments in reducing the number of entangling gates. The latter are currently considered to be a dominating source of errors in quantum computations. Some of these improvements have been made possible by using mid-circuit measurements. To our knowledge the latter feature is currently available only on the H0 quantum processor we run on.
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Benchmarking 16-element quantum search algorithms on superconducting quantum processors
Jan Gwinner, Marcin Briański, Wojciech Burkot, Łukasz Czerwiński, Vladyslav Hlembotskyi
We present experimental results on running 4-qubit unstructured search on IBM quantum processors. Our best attempt attained probability of success around 24.5%. We try several algorithms and use the most recent developments in quantum search to reduce the number of entangling gates that are currently considered the main source of errors in quantum computations. Comparing theoretical expectations of an algorithm performance with the actual data, we explore the hardware limits, showing sharp, phase-transition-like degradation of performance on quantum processors. We conclude that it is extremely important to design hardware-aware algorithms and to include any other low level optimizations on NISQ devices.
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Benchmarking 16-element quantum search algorithms on superconducting quantum processors
Jan Gwinner, Marcin Briański, Wojciech Burkot, Łukasz Czerwiński, Vladyslav Hlembotskyi
We present experimental results on running 4-qubit unstructured search on IBM quantum processors. Our best attempt attained probability of success around 24.5%. We try several algorithms and use the most recent developments in quantum search to reduce the number of entangling gates that are currently considered the main source of errors in quantum computations. Comparing theoretical expectations of an algorithm performance with the actual data, we explore the hardware limits, showing sharp, phase-transition-like degradation of performance on quantum processors. We conclude that it is extremely important to design hardware-aware algorithms and to include any other low level optimizations on NISQ devices.
Download
Benchmarking 16-element quantum search algorithms on superconducting quantum processors
Jan Gwinner, Marcin Briański, Wojciech Burkot, Łukasz Czerwiński, Vladyslav Hlembotskyi
We present experimental results on running 4-qubit unstructured search on IBM quantum processors. Our best attempt attained probability of success around 24.5%. We try several algorithms and use the most recent developments in quantum search to reduce the number of entangling gates that are currently considered the main source of errors in quantum computations. Comparing theoretical expectations of an algorithm performance with the actual data, we explore the hardware limits, showing sharp, phase-transition-like degradation of performance on quantum processors. We conclude that it is extremely important to design hardware-aware algorithms and to include any other low level optimizations on NISQ devices.
Download
Introducing Structure to Expedite Quantum Search
Marcin Briański, Jan Gwinner, Vladyslav Hlembotskyi, Witold Jarnicki, Szymon Pliś, Adam Szady
We present a novel quantum algorithm for solving the unstructured search problem with one marked element. Our algorithm allows generating quantum circuits that use asymptotically fewer additional quantum gates than the famous Grover's algorithm and may be successfully executed on NISQ devices. We prove that our algorithm is optimal in the total number of elementary gates up to a multiplicative constant. As many NP-hard problems are not in fact unstructured, we also describe the \emph{partial uncompute} technique which exploits the oracle structure and allows a significant reduction in the number of elementary gates required to find the solution. Combining these results allows us to use asymptotically smaller number of elementary gates than the Grover's algorithm in various applications, keeping the number of queries to the oracle essentially the same. We show how the results can be applied to solve hard combinatorial problems, for example Unique k-SAT. Additionally, we show how to asymptotically reduce the number of elementary gates required to solve the unstructured search problem with multiple marked elements.
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Introducing Structure to Expedite Quantum Search
Marcin Briański, Jan Gwinner, Vladyslav Hlembotskyi, Witold Jarnicki, Szymon Pliś, Adam Szady
We present a novel quantum algorithm for solving the unstructured search problem with one marked element. Our algorithm allows generating quantum circuits that use asymptotically fewer additional quantum gates than the famous Grover's algorithm and may be successfully executed on NISQ devices. We prove that our algorithm is optimal in the total number of elementary gates up to a multiplicative constant. As many NP-hard problems are not in fact unstructured, we also describe the \emph{partial uncompute} technique which exploits the oracle structure and allows a significant reduction in the number of elementary gates required to find the solution. Combining these results allows us to use asymptotically smaller number of elementary gates than the Grover's algorithm in various applications, keeping the number of queries to the oracle essentially the same. We show how the results can be applied to solve hard combinatorial problems, for example Unique k-SAT. Additionally, we show how to asymptotically reduce the number of elementary gates required to solve the unstructured search problem with multiple marked elements.
Download
Introducing Structure to Expedite Quantum Search
Marcin Briański, Jan Gwinner, Vladyslav Hlembotskyi, Witold Jarnicki, Szymon Pliś, Adam Szady
We present a novel quantum algorithm for solving the unstructured search problem with one marked element. Our algorithm allows generating quantum circuits that use asymptotically fewer additional quantum gates than the famous Grover's algorithm and may be successfully executed on NISQ devices. We prove that our algorithm is optimal in the total number of elementary gates up to a multiplicative constant. As many NP-hard problems are not in fact unstructured, we also describe the \emph{partial uncompute} technique which exploits the oracle structure and allows a significant reduction in the number of elementary gates required to find the solution. Combining these results allows us to use asymptotically smaller number of elementary gates than the Grover's algorithm in various applications, keeping the number of queries to the oracle essentially the same. We show how the results can be applied to solve hard combinatorial problems, for example Unique k-SAT. Additionally, we show how to asymptotically reduce the number of elementary gates required to solve the unstructured search problem with multiple marked elements.
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A short note on graphs with long Thomason's chains
Marcin Briański, Adam Szady
We present a family of 3-connected cubic planar Hamiltonian graphs with an exponential number of steps required by Thomason's algorithm. The base of the exponent is approximately 1.1812..., which exceeds previous results in the area.
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A short note on graphs with long Thomason's chains
Marcin Briański, Adam Szady
We present a family of 3-connected cubic planar Hamiltonian graphs with an exponential number of steps required by Thomason's algorithm. The base of the exponent is approximately 1.1812..., which exceeds previous results in the area.
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A short note on graphs with long Thomason's chains
Marcin Briański, Adam Szady
We present a family of 3-connected cubic planar Hamiltonian graphs with an exponential number of steps required by Thomason's algorithm. The base of the exponent is approximately 1.1812..., which exceeds previous results in the area.
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Links
© 2025 BEIT Inc.
Our offices
Poland:
Mogilska 43
31-545 Kraków
Canada:
101 College St
Suite H230-1
Toronto
USA:
7757 Baltimore Avenue
Ste 1603
20740 MD College Park
Links
© 2025 BEIT Inc.
Our offices
Poland:
Mogilska 43
31-545 Kraków
Canada:
101 College St
Suite H230-1
Toronto
USA:
7757 Baltimore Avenue
Ste 1603
20740 MD College Park
Links