Quantum computing system

Abstract

A quantum computing system is described, comprising a quantum processor comprising multiple qubits, the quantum processor being configured to implement one or more operations on the multiple qubits; and an energy storage system configured to store energy in a quantum state, wherein the energy storage system is coupled to the multiple qubits and configured to supply the multiple qubits with energy from the quantum state to perform the one or more operations. A method for operating a quantum processor is also described, the method comprising initialising an energy storage system into a quantum state; and performing one or more operations on multiple qubits of the quantum processor by supplying the multiple qubits with energy from the quantum state.

Description

" Quantum computing system"Cross-Reference to Related Applications

[0001] The present application claims priority from Australian Provisional Patent Application No 2025900595 filed on 28 February 2025, the contents of which are incorporated herein by reference in their entirety.Technical Field

[0002] This disclosure relates to a quantum computing system.Background

[0003] Quantum computing leverages the principles of quantum mechanics to process information to perform computations. Unlike classical computers, which use bits as the smallest unit of data (representing 0 or 1), quantum computers use qubits, which can exist in superpositions of states, enabling them to represent both 0 and 1 simultaneously. This property, along with other quantum mechanical properties such as entanglement, enables quantum computers to solve certain complex problems much more efficiently than classical computers. Potential applications of quantum computing include cryptography, optimisation, drug discovery, and solving complex mathematical problems.

[0004] Qubits are suspectable to environmental noise which may causes errors in the quantum computation. Cryogenic cooling is used in quantum computing to isolate the quantum system from the noisy environment which thermally populate the two qubit states and thus prevents any quantum computation. The implementation of quantum logic in cryogenic quantum computers requires continuous energy supply from room-temperature control electronics, leading to complex cabling, expensive control systems, and unnecessary significant heat generation.Fundamentally, unitary gates conserve entropy, implying that their operation should not generate heat. However, in practice, dependence on external energy sources limits scalability due to control channel density and heat dissipation. In particular, generating zero heat for zero entropy change is not possible in current systems due to the required attenuation along drive lines that deliver drive (XY) microwave pulses to the qubits.

[0005] Any discussion of documents, acts, materials, devices, articles or the like which has been included in the present specification is not to be taken as an admission that any or all of these matters form part of the prior art base or were common general knowledge in the field relevant to the present disclosure as it existed before the priority date of each of the appended claims.

[0006] Throughout this specification the word “comprise”, or variations such as “comprises” or “comprising”, will be understood to imply the inclusion of a stated element, integer or step, or group of elements, integers or steps, but not the exclusion of any other element, integer or step, or group of elements, integers or steps.Summary

[0007] According to an aspect of the present disclosure, there is provided a quantum computing system comprising:a quantum processor comprising multiple qubits, the quantum processor being configured to implement one or more operations on the multiple qubits; andan energy storage system configured to store energy in a quantum state,wherein the energy storage system is coupled to the multiple qubits and configured to supply the multiple qubits with energy from the quantum state to perform the one or more operations.

[0008] In some embodiments, the quantum computing system further comprises a control element configured to create entanglement between a quantum state of one of the multiple qubits and the quantum state of the energy storage system; and the quantum state of the one of the multiple qubits remains entangled with the quantum state of the energy storage system after performing one of the one or more operations on the multiple qubits.

[0009] In some embodiments, the quantum state of the energy storage system is a bosonic mode.

[0010] In some embodiments, the energy storage system is configured tocreate a qubit excitation by annihilating of quanta in the energy storage system; and annihilate the qubit excitation by creating of quanta in the energy storage system.

[0011] In some embodiments, the quantum state of the energy storage system is a Fock state.

[0012] In some embodiments, the energy storage system comprises a resonance cavity for electromagnetic waves and the multiple qubits are coupled electromagnetically to the resonance cavity.

[0013] In some embodiments, the quantum computing system further comprises control controls to tune a resonant frequency of one or more of the multiple qubits.

[0014] In some embodiments, the multiple qubits are superconducting qubits and the quantum computing system comprises flux lines configured to provide flux control of the multiple qubits.

[0015] In some embodiments, the energy storage system comprises multiple energy storage systems each associated with a subset of the multiple qubits.

[0016] In some embodiments, the energy storage system comprises a Fock state larger than or equal to a number of the multiple qubits.

[0017] In some embodiments, a ratio of a size of the quantum state of the energy storage system and a number of the multiple qubits is between 1 and 2.

[0018] According to an aspect of the present disclosure, there is provided a method for operating a quantum processor, the method comprising:initialising an energy storage system into a quantum state; andperforming one or more operations on multiple qubits of the quantum processor by supplying the multiple qubits with energy from the quantum state.

[0019] In some embodiments, performing the one or more operations comprises setting a detuning frequency corresponding to a difference between a resonant frequency of one or more of the multiple qubits and a resonant frequency of the energy storage system.

[0020] In some embodiments, performing the one or more operations comprises performing a local gate on one of the multiple qubits by performing multiple tuning steps to at least one of the multiple qubits.

[0021] In some embodiments, performing the one or more operations comprises performing an energy transfer gate on one of the multiple qubits by setting the detuning frequency of the one of the multiple qubits to about zero.

[0022] In some embodiments, performing the energy transfer gate entangles a quantum state of the one of the multiple qubits with the quantum state of the energy storage system, wherein the quantum state of the one of the multiple qubits remains entangled with the quantum state of the energy storage system after performing one of the one or more operations on the multiple qubits.

[0023] In some embodiments, performing the one or more operations comprises performing a relative phase gate by setting the detuning frequency to a different value for each of the multiple qubits.

[0024] In some embodiments, each of the one or more operations is a unitary gate performed over a gate time, wherein the unitary gate is determined based on the gate time and the detuning frequency.

[0025] In some embodiments, performing the one or more operations comprises performing an entangling gate on a subset of multiple qubits by setting the detuning frequency for each of the subset to a similar value.

[0026] In some embodiments, performing the one or more operations comprises performing a collective gate on the multiple qubits by performing one or more detuning steps for each of the multiple qubits.

[0027] According to an aspect of the present disclosure, there is provided a method for determining an ancilla qubit state in relation to multiple data qubits, the method comprising: tuning an ancilla qubit to transfer energy from a quantum state of an energy storage system, to entangle the ancilla qubit with the quantum state of the energy storage system resulting in a changed quantum state of the energy storage system;tuning the multiple data qubits to transfer energy between the multiple data qubits, wherein the transfer energy between the multiple data qubits is based on the changed quantum state of the energy storage system to create a relationship between the ancilla qubit state and a parity of the multiple data qubits;tuning the ancilla qubit to transfer energy between the ancilla qubit and the energy storage system with a direction based on the parity of the multiple data qubits; and measuring the ancilla qubit to determine the ancilla qubit state, the ancilla qubit state reflecting the parity of the multiple data qubits.

[0028] In some embodiments, tuning the ancilla qubit comprises tuning the ancilla qubit to disentangle the ancilla qubit from the changed quantum state of the energy storage system resulting in a pure quantum state of the ancilla qubit that reflects the parity of the multiple data qubits.

[0029] In some embodiments, the method further comprises, after tuning the ancilla qubit to disentangle the ancilla qubit, further tuning the multiple data qubits to restore an original quantum state of the multiple data qubits.

[0030] In some embodiments, the method further comprises performing quantum error correction based on the determined ancilla qubit state.

[0031] According to an aspect of the present disclosure, there is provided a method for creating a quantum error correction code logical qubit on a quantum computing system comprising an energy storage system, the method comprising tuning a resonant frequency of multiple qubits of a quantum processor using the energy storage system to entangle the multiple qubits into the logical qubit.

[0032] In some embodiments, the method further comprises performing a probing operation, wherein the quantum error correction code logical qubit is in a superposition state after the probing operation, and performing the probing operation comprises performing a controlled-parity operation of a subset of the multiple qubits using an ancilla qubit of the quantum processor.

[0033] In some embodiments, the method further comprisesprobing a stabilizer operation using a controlled-parity operation; andperforming error correction of the quantum error correction code logical qubit based on a result of probing the stabilizer operation.

[0034] Optional features provided in relation to the quantum computing system, may equally apply as optional features to the methods described above.Brief Description of Drawings

[0035] An example will be described with reference to the following drawings:

[0036] Fig. 1 illustrates an example embodiment of a quantum computing system.

[0037] Fig. 2 illustrates another example embodiment of a quantum computing system.

[0038] Fig. 3 illustrates an example method of operating a quantum processor.

[0039] Fig. 4 illustrates an example embodiment of a quantum computing system comprising a quantum processor with multiple data qubits and an ancilla qubit.

[0040] Fig. 5 illustrates an example method for determining an ancilla qubit state in relation to multiple data qubits.

[0041] Fig. 6 illustrates an example circuit for determining an ancilla qubit state in relation to multiple data qubits.

[0042] Fig. 7 illustrates an example method of creating a logical qubit for quantum error correction on a quantum computing system comprising an energy storage system.

[0043] Fig. 8 illustrates an example circuit for creating a quantum error correction code logical qubit on a quantum computing system comprising an energy storage system.

[0044] Fig. 9 illustrates an example circuit for creating a superposition state of a quantum error correction code logical qubit on a quantum computing system comprising an energy storage system.

[0045] Fig. 10 illustrates an example embodiment of the disclosed quantum computing system.

[0046] Fig. 11 illustrates a Hamiltonian of the quantum computing system, showing a block diagonal structure of computational subspaces.

[0047] Fig. 12a shows a collective energy -transfer gate time versus the number of qubits for different ratios of battery quantum per qubit.

[0048] Fig. 12b shows a collective energy -transfer gate error versus battery quantum per qubit.

[0049] Fig. 13 illustrates an 0(1) multi-qubit parity probing protocol according to the present disclosure.

[0050] Fig. 14 illustrates another example of multi-qubit parity probing according to the present disclosure.

[0051] Fig. 15 illustrates encoding a quantum error correction (QEC) logical state.

[0052] Fig. 16 illustrates computation of a three-qubit circuit according to the present disclosure.

[0053] Fig. 17 shows a heat analysis and scaling opportunities of the shared-cavity quantum computation.

[0054] Fig. 18 illustrates an effect of including counter-rotating terms in the Hamiltonian of the quantum computing system.

[0055] Fig. 19 shows results of heat power source distribution for the quantum computing system.

[0056] Fig. 20 shows fidelity of non-Pauli energy-exchange gates as a function of the total number of initial excitations in the energy storage systems for different multi -qubit systems.Description of Embodiments

[0057] Disclosed herein are a quantum computing system and methods relating to such system. The quantum computing system according to the present disclosure comprises an energy storage system that stores energy in a quantum state and supplies energy to the multiple qubits of the quantum computing system to perform one or more operations (such as quantum computing operations). As such, a quantum computation may be performed on the multiple qubits without using drive lines for qubit control. More specifically, all qubits are connected to a shared energysystem, eliminating the need for individual qubit drive lines and their associated attenuators which generate active and passive heat during computation.

[0058] By eliminating the need for a drive line for each qubit, the disclosed quantum computing system reduces energy consumption to readout-only and increases the number of qubits per cryogenic system by a factor of four. More specifically, the results discussed herein show that the disclosed quantum computing system reduces active heat generation and enables above 20% (or 4-fold with superconducting cables) increase in qubits per cryogenic fridge. It is also demonstrated herein that increasing the number of qubits, while maintaining fixed size of the quantum state of the energy storage system (e.g., a fixed quanta) per qubit, enhances gate fidelity and speed superextensively.

[0059] As previously discussed, unitary gates preserve entropy, implying that their operation should not generate heat, but in practice, quantum computers generate active heat when executing unitary gates. However, the disclosed quantum computing system could potentially reach the fundamental limit of zero active heat generation during unitary logic. The von Neumann entropy of a quantum system remains unchanged under any unitary transformation and is zero for any pure state. As a result, since entropy does not change during a unitary quantum circuit, computation can, in principle, proceed without generating heat, as dictated by the thermodynamic bound ΔQ ≥ -TΔS, where ΔQ is heat generation, ΔS, is entropy change, and T is temperature. Achieving this bound precisely — specifically, generating zero heat for zero entropy change — is impossible with external drive pulses due to attenuation losses. In addition, forcing separability between the energy storage system and the qubits also mathematically introduce gate errors which increase the entropy of the qubit system. However, the disclosed quantum computing system leverages quantum field recycling and the entanglement between the energy storage system and the qubit, enabling it to approach the thermodynamic limit. In other words, using a closed system of this disclosure and enabling entanglement between any one of the multiple qubits and the energy storage system, may approach the limit by reusing the energy charged into the energy storage system prior to the circuit. This highlights the potential using such energy storage systems to enable energy -efficient quantum computation and to move towards thermodynamically optimal quantum processing. Practically, this limit can be reached with superconducting flux lines that may run quantum information processing in the disclosed quantum computing system without generating heat. As such, the disclosed quantum computing system can enable energy -efficient quantum computation and thermodynamically optimal quantum processing.

[0060] In this disclosure, an energy storage system (configured to store energy in a quantum state) is used as an energy source for quantum computation that facilitates unitary logic and can provide a universal gate set. Unlike classical power sources, the energy storage system according to the present disclosure maintains quantum coherence with its load. The energy storage system of the present disclosure may be considered to be a quantum battery, which is a type of battery that uses the principles of quantum mechanics to improve charging, storage and release of energy. Quantum batteries may be defined as ^ / -dimensional systems which store energy in excited states, support quantum coherence between its energy states or with its charger or load and enable reversible charging and work extraction through unitary operations.

[0061] In some embodiments of the quantum computing system, coupling the energy storage system to the multiple qubits of a quantum processor enables a universal quantum gate-set with a single controlling parameter, specifically the qubits’ resonant frequency. For example, when tuned to resonance, the energy storage system may facilitate energy-transfer (XY) gates, showing the energy storage system’s superextensive hallmarks where increasing the number of qubits reduces charging gate times and enhances their fidelities. Further, multi-qubit parity probing protocol may be achieved with a single entangling gate and simulate the encoding of a d = 2 surface code logical-X state with >98% fidelity.

[0062] In some embodiments, initialising the energy storage system in a quantum state may supply the energy required for arbitrary unitary gates regardless of the circuit’s depth, via the recycling of pre-charged energy. This may enable entanglement between the quantum state of the energy storage system and the quantum state of any of the multiple qubits during computation. As a result, this may lower the initial energy requirements of the energy storage system below energy-fidelity bounds. Entanglement between the quantum state of the energy storage system and the qubit states also has other advantages, as will be discussed in this disclosure.

[0063] For example, this entanglement addresses the limitation of having the energy storage system and the qubits in a separable quantum state at the end of a computation. Such cases may force separability to keep qubits’ state pure. In such cases, the qubit and the energy storage system may be coupled, but this interaction is fixed to specific cases that excluding entangling. This may cause extra errors during computation. As such, having the energy storage system and the qubits in a separable quantum state at the end of a computation can reduce gate fidelities e.g., increase error rate of computed quantum gates, and may require more energy. In contrast, theentanglement disclosed herein provides a more general interaction between the qubits and the energy storage system. For example, after entanglement, the entanglement between any one of the multiple qubits and the quantum state of the energy storage system may be preserved or maintained, even after subsequent operations on any one of the multiple qubits. This is in contrast to other cases where separability of the qubit states is forced. Further, simulation showed almost perfect gate fidelities when the quantum energy storage system had significantly less energy than the bounds possible by having the energy storage system and the qubits in a separable quantum state at the end of a computation.Quantum computing system

[0064] Fig. 1 illustrates an example embodiment of a quantum computing system (denoted as quantum computing system 100). Fig. 1 is one example of a configuration of system 100.However, system 100 is not strictly limited to this configuration and this may be one possible embodiment of system 100. It is noted that system 100 of Fig. 1 is only meant to illustrate an example system.

[0065] Quantum computing system 100 comprises quantum processor 110 comprising multiple qubits 111. In this example, multiple qubits 111 are in the form of a 3-by-3 array of qubits but other arrangements including linear arrays and other structures with many more qubits are equally possible. A wide range of physical platforms can be used, including superconducting qubits, cold atoms, trapped ions, quantum dots, nitrogen vacancies,, phosphorous in silicon and others. Multiple qubits 111 may be superconducting qubits made from materials like aluminium and niobium, trapped ion qubits using ions such as calcium and ytterbium, and semiconductor qubits based on silicon and gallium arsenide. Multiple qubits 111 may also be topological qubits made from exotic materials like topological insulators and superconductors.

[0066] It is noted that each of multiple qubits 111 may have a resonant frequency, depending on the specific type of qubits and its physical implementation. In some examples, each of multiple qubits 111 may have a different or similar resonant frequency. The resonant frequency of multiple qubits 111 may be in the microwave range. The resonant frequency of a qubit is the specific frequency at which the qubit transitions between its two energy states, typically denoted as the ground state |0) and the excited state |1). In this disclosure, the resonant frequency may also be referred as qubit energy. As will be discussed later in this disclosure, the resonant frequency of any one of multiple qubits 111 may be “tuned”, such that the resonant frequency ofany one of multiple qubits 111 changes (e.g., the energy between the ground state |0) and the excited state |1) changes). Tuning the resonant frequency of multiple qubits 111 may be performed using control controls (such as flux or superconducting control lines), lasers, or other suitable equipment implemented in quantum computing system 100. “Tuning” may refer generally to adjusting the resonant frequency which may be towards an external field to achieve coupling or away from the external field. In a case where the qubit is resonant with the external field (e.g., the frequencies match) and the resonant frequency is tuned to reduce the coupling, this could be referred to as “detuning” which is also a type of tuning in this context. This is not to be confused with detuning in the meaning of loss of quantum information to the external environment due to noise, also referred to as dephasing.

[0067] Quantum processor 110 is configured to implement one or more operations on multiple qubits 111. For example, the one or more operations may be gate operations, which may be referred to as quantum gates, unitary gates, unitary operators, Clifford gates, non-Clifford gates or the like. As will be further discussed, the quantum computing system according to the present disclosure supports all unitary gates and hence, quantum processor 110 may be configured to implement any unitary gates including single qubit gates (which may be referred to as local gates) and multi-qubit gates (which may be referred to as non-local gates). Single qubit gates may include, but are not limited to, the Pauli-X gate (or simply the X gate), the Hadamard gate and the phase shift gate. The multi -qubit gates may include, but are not limited to, the entangling gates such as CNOT (Controlled-NOT), Toffoli (Controlled-Controlled-NOT), or ISWAP. Quantum processor 110 may be configured to implement other operations on multiple qubits, such as, but not limited to, measurement.

[0068] Quantum computing system 100 comprises energy storage system 120 configured to store energy in a quantum state (such as quantum state 121). It is noted that energy storage system 120 may also be configured to receive energy (in the form of quantum, for example) from multiple qubits 111. In this sense, energy storage system 120 may be configured to perform reserve (or reversible) energy transfer. In some examples, quantum state 121 of energy storage system 120 may be a fermionic quantum state comprised of one or multiple particles In some examples, energy storage system 120 stores (or contains) the energy in an excited state of a bosonic single or multiple modes such as a quantum harmonic oscillator, such as a mechanical drum, lattice movements (phonons), photons, photonic-quasiparticles (such as plasmon polaritons, phonon polaritons, or exciton polaritons) etc. In general, quantum state 121 may be a quantum state of multiple quanta (i.e., comprised of multiple particles or multiple excitations).However, in other examples, quantum state 121 may be a quantum state of a single particle or single excitation.

[0069] In some examples, energy storage system 120 may be molecules in optical cavities, superconducting qubits or a nuclear magnetic resonance (NMR) spin system. Moreover, energy storage system 120 may be a high- cavity mode, a many -body quantum system or an open quantum system. In essence, energy storage system 120 may be a quantum battery. As will be described below, in some embodiments, energy storage system 120 may be a Fock state quantum battery. However, in other embodiments, energy storage system 120 may be a coherent state quantum battery.

[0070] A notable difference between energy storage system 120 (which may be considered to be a quantum energy storage system) and a classical energy storage system lies in the coherence between quantum energy states. The quantum coherence facilitates collective effects as Dicke superradiance and its inverse, superabsorption, enabling global energy transfer between the battery and load, enabling the energy transfer rate, or charging / discharging power, to scale superextensively. For example, if energy storage system 120 comprised a quantum state of multiple quanta, as the number of quanta (or load quanta) N increases, the charging (or discharging) power per quanta scales as N. Additionally, the reversible transfer of energy to and from energy storage system 120 enables them to surpass the Landauer energy-cost limit, making them advantageous for quantum computing. Moreover, a notable characteristic of energy storage systems of the present disclosure is their superextensive speed-up in energy transfer.

[0071] Energy storage system 120 is coupled to multiple qubits 111 of quantum processor 110. “Coupled” in the context of the present disclosure may refer to multiple qubits 111 and energy storage system 120 being linked, connected or the like, which enables multiple qubits 111 and energy storage system 120 to interact to transfer energy between energy storage system 120 and multiple qubits 111. In some examples, energy storage system 120 may be physically connected to quantum processor 110 or multiple qubits 111. Multiple qubits 111 may also be within energy storage system 120. Energy storage system 120 may be coupled to multiple qubits 111 using a modulator or the like. For example, a modulated coupler may affectively tune the difference in the resonant frequency of multiple qubits 111 compared to the resonant frequency of energy storage system 120. Quantum computing system 100 may comprise one or more control lines configured to couple one of multiple qubits 111 with energy storage system 120.

[0072] Energy storage system 120 is also configured to supply multiple qubits 111 with energy from quantum state 121 to perform the one or more operations on multiple qubits 111. For example, quantum state 121 of energy storage system 120 may correspond to a state of multiple quanta, and energy storage system 120 may supply energy to one of multiple qubits 111 by supply the qubit with a quantum. This may therefore change (e.g., increase) the energy of that qubit. Inversely, multiple qubits 111 may be configured to supply energy (in the form of quanta, for example) to quantum state 121. Supplying multiple qubits 111 with energy from quantum state 121 (and vice versa) may occur via a Tavis-Cummings interaction.

[0073] In some embodiments, quantum state 121 of energy storage system 120 may be a quantum state of multiple quanta. As such, in some embodiments, energy storage system 120 is configured to create a qubit excitation by annihilating of quanta in energy storage system 120; and annihilate the qubit excitation by creating of quanta in energy storage system 120. “Qubit excitation” may also be referred to as an excited qubit or a qubit in an excited state (i.e., 11)). “Qubit excitation” may also be referred to as a dressed qubit, i.e., a qubit that has been dressed by a quantum of energy storage system 120. Given this embodiment, it is noted that the combined number of quanta in energy storage system 120 and qubit excitations is conserved when performing one or more operations on multiple qubits 111. It is also possible that quantum state 121 of energy storage system 120 is a superposition, such as a superposition of number states.

[0074] In some embodiments, energy storage system 120 comprises a cavity and multiple qubits 111 are coupled to the cavity. In general, “cavity” may refer to a space within a solid object which can store electromagnetic fields. In the context of the present disclosure, the cavity may store (or contain) quantum state 121. In some examples, multiple qubits 111 may be within the cavity to couple multiple qubits 111 to energy storage system 120. However, in other examples, multiple qubits 111 may be external to the cavity. In some embodiments, energy storage system 120 comprises a resonance cavity for electromagnetic waves and multiple qubits 111 are coupled to electromagnetically the resonance cavity. A resonant cavity is a structure that confines electromagnetic waves and enables them to resonate at specific frequencies. As such, quantum state 121 may be resonant at a resonant frequency. In some embodiments, energy storage system 120 may also be an optical cavity.

[0075] In some embodiments, quantum state 121 is a bosonic mode or a fermionic mode. “Bosonic mode” may refer to quantum state 121 obeying Bose-Einstein statistics. Similarly,“fermionic mode” may refer to quantum state 121 obeying Fermi-Dirac statistics. In one example, quantum state 121 may be a quantum state of multiple bosons (e.g., photons, phonon etc). More specifically, quantum state 121 may comprise multiple quanta where these quanta are bosons. In a specific example, in a cavity filled with electromagnetic radiation, the different possible standing wave patterns of the electromagnetic field are the bosonic modes of the field. Each mode can be occupied by any number of photons, which are bosons. However, in other embodiments, quantum state 121 is a fermionic mode. For example, quantum state 121 may be a quantum state of multiple fermions, such as electrons.

[0076] In some embodiments, the quantum state is a Fock state. A Fock state, which may be referred to as number states or Fock number states is a quantum state that is an element of a Fock space with a single well-defined number of particles. This disclosure generally refers to the use of Fock states as single-mode Fock states | n }, where n is the number of particles that the Fock state describes. In some embodiments, energy storage system 120 may be a bosonic mode in a Fock state. For example, quantum state 121 is a quantum state of multiple photons and hence, Fock states represent photon number states. As such, in some embodiments, quantum state 121 of energy storage system 120 is a photon number state. As will be discussed later in the disclosure, having quantum state 121 of energy storage system 120 as a photon number state (i.e., a quantum state of light) may be advantageous in overcoming previous bounds on the gate fidelities and minimal energy requirements.

[0077] However, in other embodiments, quantum state 121 is a coherent state. Coherent states are specific types of quantum states that exhibit classical-like behaviour while still retaining quantum properties. More specifically, coherent states | a ) are eigenstates of the annihilation operator a: a | a )= a | a ), which can be created by applying the displacement operator D(a) to the vacuum state which displaces the vacuum state from the origin in phase space by amount a. The vacuum state is a coherent state with zero amplitude. Coherent states are minimum uncertainty states which satisfy the lower bound of the uncertainty principle.

[0078] In some embodiments, quantum computing system 100 further comprises control lines to tune a resonant frequency of one or more of the multiple qubits. In some examples, the control lines may be flux lines which modify the electromagnetic flux within a superconducting circuit. For example, in one embodiment, multiple qubits 111 are superconducting qubits and quantum computing system 100 comprises flux lines configured to provide flux control of the multiplequbits. The flux lines may be comprised from superconducting materials. Relying solely on such control lines, rather than drive control lines, may be advantageous in reducing energy consumption. In particular, these control lines do not include attenuators in the lowest cryogenic layer and therefore do not dissipate substantial energy, which is in contrast to drive lines which use attenuators for isolating the qubits system from thermal noise and hence dissipate energy.

[0079] As previously discussed, quantum state 121 may be a bosonic mode and in some examples, quantum state 121 may be a bosonic mode in a Fock state. In some embodiments, the Fock state is larger than or equal to a number of multiple qubits 111. More specifically, as the Fock state is a number state of multiple quanta, in these embodiments, the number of quanta that make up the Fock state may be larger than or equal to the number of multiple qubits 111. As will be discussed later in the disclosure, it may be an advantage for the Fock state of energy storage system 120 to be larger or equal to the number of multiple qubits 111 as the unitary gate charging time approaches the superextensive limit of l / / , thereby reducing the gate time to the superextensive limit.

[0080] In some embodiments, a ratio of a size of quantum state 121 (such as the number of quanta) of energy storage system 120 and a number of multiple qubits 111 is between 1 and 2. As such, the size of quantum state 121 is larger than the number of multiple qubits 111.However, although a large sized quantum state may be useful, low-error unitary computation may be achieved even when this ratio of below 2. As such, a ratio of between 1 and 2 may provide a useful trade-off between error and difficulty in creating quantum state 121.

[0081] In some embodiments, quantum computing system 100 further comprises a control element configured to create entanglement between quantum state of one of multiple qubits 111 (or at least one of multiple qubits 111) with quantum state 121 of energy storage system 120. For example, the quantum state of the one of multiple qubits 111 (which may be referred to as a qubit state) may be entangled with quantum state 121 by performing an energy transfer gate on the one of multiple qubits 111. In some examples, the quantum state of energy storage system 120 (i.e., quantum state 121) is a Fock state | n } and the energy transfer gate may entangle the qubit state with the Fock state to produce the quantum state: | 0)® | n )+ 11)® | n -1 ). As such, the quantum state of the one of multiple qubits 111 is dependent on quantum state 121 of energy storage system 120. In other words, the quantum state of the one of multiple qubits Ill is dependent on energy storage system 120 transferring a quanta to the one of multiple qubits 111.

[0082] The control element may be a flux line, drive line, a laser, a pulse generator, or another device configured to create entanglement. For example, the control element may be a capacitor located between the one of multiple qubits 111 and energy storage system 120, which enables an electron to ‘jump’ between the one of multiple qubits 111 and energy storage system 120.

[0083] Entangling a qubit state with quantum state 121 of energy storage system 120 has many advantages. For example, as will be further discussed in this disclosure, the simulations of such entanglement show almost perfect gate fidelities when energy storage system 120 has significantly less energy compared to other quantum energy storage systems. Further, such entanglement enables a probing operation (such as a controlled-parity operation) to be performed, which may be used to create error correcting codes. Even further, such entanglement may significantly reduce the total circuit time compared to other quantum gate computers. More specifically, such entanglement may significantly reduce the total circuit time compared to using sequence of single-qubit and pairwise entangling gates on a quantum gate computer.

[0084] Moreover, in some embodiments, the quantum state of the one of the multiple qubits remains entangled with quantum state 121 of energy storage system 120 after performing one of the one or more operations on the multiple qubits. More specifically, the control element may create entanglement between a quantum state of one of multiple qubits 111 and quantum state 121 of energy storage system 120 and then one or more operations (i.e., quantum gates) may be applied to any number of multiple qubits 111. However, the entanglement may remain intact before and after performing such operations. This is in contrast to a system which forces separability of quantum states and may destroy the entanglement after performing one or more operations. Maintaining the entanglement is useful as this enables the ability to execute many different quantum circuits. For example, as will be discussed later in the disclosure, such entanglement between a quantum state of one of multiple qubits 111 and quantum state 121 of energy storage system 120 enables a probing operation to be established, thereby enabling the parity of multiple qubits 111 to be determined.

[0085] Fig. 2 illustrates another example embodiment of a quantum computing system (denoted as quantum computing system 200). Energy storage system 220 comprises multiple energy storage systems (energy storage system 221, 222, 223). Each of the energy storage system 221, 222, 223 is associated with a subset of the multiple qubits of quantum processor 210. For example, subset 211 is associated with energy storage system 221. In other words, the individual energy storage systems may be coupled to an associated subset of the multiple qubits.For example, energy storage system 221 is coupled to subset 211. As such, each individual energy storage system may supply energy to their associated subset of multiple qubits. Such an embodiment may be advantageous so that the quantum state of energy storage system 220 does not have to be large. For example, energy storage system 221, 222, 223 may store energy as a quantum state, where the quantum state is a bosonic mode of a Fock state. As such, having multiple energy storage system means that multiple smaller Fock state may be used rather than a single large Fock state. Such a quantum computing system may be easier to construct and less resource intense as a large quantum state (e.g., a large Fock state) may not be used to supply energy to the multiple qubits.A method for operating a quantum computing system

[0086] Fig. 3 illustrates an example method (denoted as method 300) of operating a quantum processor (such as quantum processor 110) according to an aspect of the present disclosure. Method 300 will be discussed with reference to Fig. 1, and some embodiments of quantum computing system 100 described above may equally be embodiments of method 300.

[0087] Method 300 comprises initialising 301 energy storage system 120 into quantum state 121. Initialising 301 energy storage system 120 may refer to creating quantum state 121 by powering energy storage system 120, for example. Initialising 301 energy storage system 120 may refer to providing energy storage system 120 and creating quantum state 121. Initialising 301 energy storage system 120 may refer to providing energy storage system 120 with an established (or previously created) quantum state 121. Initialising 301 energy storage system 120 may also refer to coupling energy storage system 120 with multiple qubits 111 of quantum processor 110, if not already coupled. In essence, initialising 301 energy storage system 120 refers to “setting up” energy storage system 120 so that one or more operations may be performed on multiple qubits 111.

[0088] For example, quantum state 121 may be a bosonic mode in a Fock state. In this example, the Fock state may be a photon number state and hence, the quanta of the Fock state are photons. In this example, initialising 301 energy storage system 120 may comprise generating a Fock state using strong laser beam (e.g., a “pump” beam) directed at a non-linear crystal (such as a beta-barium borate or lithium niobate crystal). For example, this interaction between the strong laser bean and the non-linear crystal may produce a pair of photons through the process of spontaneous parametric down-conversion, where one of the photons in the pair ismeasured, such that the measurement collapses the other photon into a Fock state. This is just one example of the generation of a Fock state. However, other generation methods may be equally applicable, for example using a microwave pulse generator and a mediator transmon qubit as the non-linear element.

[0089] Method 300 comprises performing 302 one or more operations on multiple qubits 111 of quantum processor 110 by supplying multiple qubits 111 with energy from quantum state 121. As previously discussed, energy storage system 120 may supply multiple qubits 111 with energy from quantum state 121 by providing (e.g., exchanging) a quantum (such as a photon, in some examples) between quantum state 121 and one of multiple qubits 111. Supplying energy from energy storage system 120 to multiple qubits 111 may cause a unitary gate operation to be performed on multiple qubits 111 and hence, supplying energy from energy storage system 120 to multiple qubits 111 may perform a quantum computation.

[0090] In some embodiments, performing 302 the one or more operations comprises setting a detuning frequency corresponding to a difference between a resonant frequency of one or more of multiple qubits 111 and a resonant frequency of energy storage system 120. It is noted here that the term resonance or resonant may refer to the matching of energy levels. For example, the qubit may be tuned so that the energy difference (i.e., gap) between the qubit states matches the energy between quanta of the energy supply system to facilitate energy transfer between the qubit and the energy supply system. This may be referred to as resonance. So, the detuning frequency may simply resemble an energy difference between one or more of multiple qubits 111 and the energy of energy storage system 120 (i.e., the energy of the quanta at resonance).

[0091] The detuning frequency may be referred to as a detuning value, detuning energy or simply as detuning to create an energy difference between the energy levels. In other words, quantum computation may be performed by controlling the detuning between the qubit’s resonant frequency (e.g., the qubit’s energy) and the resonant frequency (i.e., energy level) of energy storage system 120 (e.g., the resonance of energy storage system 120). As such, initialising 301 energy storage system 120 into quantum state 121 enables a universal quantum gate-set with a single controlling parameter per qubit, namely the qubits’ resonant frequency (i.e., energy gap between qubit quantum states).

[0092] In some embodiments, performing 302 the one or more operations comprises performing an entangling gate on a subset of multiple qubits 111 by setting the detuningfrequency for each of the subset to a similar value. In other words, entanglement between qubits may be achieved when all qubits to be entangled are detuned to the same energy (i.e., entanglement gate is performed when the qubits are detuned from energy storage system 120 resonance by a similar frequency). It is noted that the entangling gate may be based on the initial state of quantum state 121 (i.e., the initial state modifies the executed entangling gate). The initial state may be a ground state, i.e., energy storage system 120 is in its vacuum state, or be a fully charged state, i.e., energy storage system 120 has not exchanged any quanta with multiple qubits 111.

[0093] Further, the system can perform local single-qubit gates without disturbing the other qubits. In the quantum computing system disclosed herein, an arbitrary Z rotation can be implemented by detuning the resonance of a single qubit, which implies that reaching a single local non-Pauli energy -changing gate is sufficient to complete the full quantum universal gate set. Such a gate may be a Hadamard, aor any non-trivial rotation that modifies the qubit’s energy.

[0094] As will be discussed in the results section, it is demonstrated that a local non-trivial energy-changing gate that produces the same unitary regardless of the states of the other qubits can be implemented with few detuning steps. Therefore, in some embodiments, performing 302 the quantum operation comprises performing a local gate (i.e., a single unitary gate operation) on one of multiple qubits 111 by performing multiple tuning steps to at least one of multiple qubits 111. For example, having up to five qubits and an energy storage system fully charged with seven photons, two detuning steps may be sufficient to achieve average gate fidelities of above 99.8% and worst-case fidelities of 99.5%. Incorporating additional detuning steps increases the available degrees of freedom, which can be exploited to further enhance fidelities and accommodate larger systems.

[0095] In some embodiments, performing 302 the one or more operations comprises performing an energy transfer gate by setting the detuning frequency to about zero. In other words, energy transfer gates may be performed for resonant energy exchange between qubits and energy storage system 120 and, as such, the detuning frequency may be set to below the coupling strength of multiple qubits 111 and energy storage system 120. In some examples, an energy transfer gate may be performed by setting the detuning frequency to about zero and setting the detuning frequency up to around 100 times the coupling strength of multiple qubits 111 and energy storage system 120. In some embodiments, performing the energy transfer gate entanglesa quantum state of the one of multiple qubits 111 with the quantum state 121 of energy storage system 120. As previously described, the quantum state of the one of multiple qubits 111 may remain entangled with quantum state 121 of energy storage system 120 after performing one of the one or more operations on multiple qubits 111.

[0096] In some embodiments, performing 302 the one or more operations comprises performing a relative phase gate, for example a local S gate or T gate, by setting the detuning frequency to a different value for each of the multiple qubits. In other words, relative-phase gates are implemented by choosing slightly different detuning energies for each qubit, ensuring that all qubits are off-resonance from one another.

[0097] In some embodiments, each of the one or more operations is a unitary gate performed over a gate time. Gate time in this context refers to the time for which the qubit is tuned to a specific energy in relation to the energy storage system (i.e., the time for which detuning is applied for). In other words, the operation (i.e., the quantum computing operation) may be performed for a certain time period or duration. In these embodiments, the unitary gate is determined based on the gate time and the detuning frequency. In other words, a certain operation (e.g., entangling or relative phase) may be performed based on the detuning frequency. However, the specific operation may be based on the gate time. For example, setting the detuning frequency for each of a subset of multiple qubits 111 to a similar value may perform an entangling operation on the subset. However, if the operation is performed for a specific gate time, the entangling operation may be an iswap gate or an iswap gate. More specifically, an 71 iswap gate or an iswap gate may be performed when the gate is applied for t = and t =2g2 / Δ71respectively, wherein A is the detuning frequency (e.g., detuning value) and g is the 452 / A’coupling constant between multiple qubits 111 and energy storage system 120. In another example, energy storage system 120 may perform an X gate when setting Aj = 0 for all of multiple qubits 111 and the operation is applied for t = — j=, where \nex) is the state of theinitial quantum state of energy storage system 120 (e.g., the number of quanta).

[0098] In some embodiments, performing the one or more operations comprises performing a collective gate on multiple qubits 111 by performing one or more detuning steps for each of multiple qubits 111. A “collective gate” may refer to a gate that is effectively multiple single qubit gate acting (effectively) simultaneously. For example, a collective X-gate may refer to anX-gate being effectively performed on each of multiple qubits 111. As will be discussed in the results section, a surprising effect of these collective gates is that increasing the number of qubits also improves gate fidelity. A perfect collective gate may be achievable when using an energy storage system in a Fock state and a single qubit with a single detuning step, as well as a perfect charging of N qubits which are charged in N detuning steps.A method for determining an ancilla state

[0099] Fig. 4 illustrates an example embodiment of a quantum computing system (denoted as quantum computing system 400), which may be similar to the example embodiment described with reference to Fig. 1. Quantum computing system 400 comprises multiple qubits (5 qubits shown in this example), where the first 4 qubits may be considered to be data qubits (denoted as multiple data qubits 411) and the fifth qubit may be considered to be an ancilla qubit (denoted as ancilla qubit 412). However, it is noted that quantum processor 410 may have any number of data qubits and ancilla qubits. In this disclosure, an ancilla qubit may also be referred to as an ancillary qubit. Multiple data qubits 411 store the data that is to be used in the computation (i.e., payload), while ancilla qubit 412 is used for probing or encoding purposes and is otherwise not used in the quantum computation. For example, ancilla qubit 412 may be used (for quantum error correction, for example) so that a measurement does not need to be performed on multiple data qubits 411, thereby collapse their individual quantum state.

[0100] Fig. 5 illustrates an example method (denoted as method 500) for determining an ancilla qubit state in relation to multiple data qubits (such as data qubit 411). Method 500 will be discussed with reference to Fig. 4. It is noted that some embodiments of quantum computing system 100 described above may equally be embodiments of method 500. Method 500 may also be considered to be a method for determining a parity between multiple qubits.

[0101] Method 500 comprises tuning 501 ancilla qubit 412 to transfer energy from quantum state 421 of energy storage system 420, to entangle ancilla qubit 412 with quantum state 421 resulting in a changed quantum state of energy storage system 420. As such, after tuning 501, ancilla qubit 412 and quantum state 421 may be in an entangled state. More specifically, ancilla qubit 412 and quantum state 421 may be in a superposition state of qubit states and quantum states of energy storage system 420. In a sense, the superposition state may be a superposition of a state where energy has been exchanged from quantum state 421 to ancilla qubit 412 and a state where no exchange has occurred. For example, if quantum state 421 is a bosonic mode of a Fockstate (i.e., a quantum state comprising multiple quanta), ancilla qubit 412 and quantum state 421 may be in a superposition state of a state comprising the ground state of ancilla qubit 412 and the initial quantum state of energy storage system 420 (i.e., a full number state) and a state comprising the excited state of ancilla qubit 412 and a different quantum state of energy storage system 420 that contains one less quantum. Similar examples apply to other entanglement between quantum state described in this disclosure. It is noted that the “changed quantum state” of energy storage system 420 may refer to the entangled state. Tuning 501 ancilla qubit 412 may comprise setting the detuning frequency to about zero in one or multiple detuning steps, as previously discussed. As such, tuning 501 ancilla qubit 412 may be considered to be applying (or performing) an energy transfer gate on ancilla qubit 412.

[0102] Method 500 comprises tuning 502 multiple data qubits 411 to transfer energy between multiple data qubits 411. Tuning 502 may entangle multiple data qubits 411 with each other. For example, tuning 502 multiple data qubits 411 may comprise performing an entangling gate on multiple data qubits 411 by setting the detuning frequency for each of multiple data qubits 411 to a similar value, as previously described. Hence, multiple data qubits 411 may exchange quanta between themselves as a result of the tuning 502. Thus, multiple data qubits 411 may become entangled with one another as a result of the tuning 502.

[0103] During tuning 502 of multiple data qubits 411, the transfer energy between multiple data qubits 411 is based on the changed quantum state of energy storage system 420, which creates a relationship between the ancilla qubit state of ancilla qubit 412 and the parity of multiple data qubits 411. As this may create entanglement between multiple data qubits 411 and since the entanglement is based on the changed quantum state of energy storage system 420, energy storage system 420 (and ancilla qubit 412) receives a phase kickback based on the parity of multiple data qubits 411 (or part thereof). Hence, there is a relationship between ancilla qubit 412 (or the ancilla qubit state, more specifically) and the parity of multiple data qubits 411. Parity probing is often a building block used in error detection and correction. For example, in quantum error correction codes, parity checks define the quantum error correction code, and their probing removes entropy from the system, and help identify and correct errors that may occur during quantum computations. Therefore, due to the relationship, ancilla qubit 412 may be used to determine a parity of multiple data qubits 411.

[0104] Method 500 comprises tuning 503 ancilla qubit 412 to transfer energy between ancilla qubit 412 and energy storage system 420 with a direction based on the parity of multiple dataqubits 411. In this context, “direction” refers to either one of: (i) transferring from ancilla qubit 412 to energy storage system 420; and (ii) transferring from energy storage system 420 to ancilla qubit 412. In some embodiments, tuning 503 produces a pure quantum state of ancilla qubit 412, wherein the pure quantum state reflects the parity of multiple data qubits 411. In other words, method 500 comprises tuning 503 ancilla qubit 412 to change quantum state of both energy storage system 420 and ancilla qubit 412 to a measurable state that reflects the parity of multiple data qubits 411. In this context, the quantum state of the ancilla qubit 412 may refer to either a |0) (e.g., ground) or |1) (e.g., excited) state. Tuning 503 ancilla qubit 412 may be performed similar to tuning 501, in which the detuning frequency may to set to about zero. As such, ancilla qubit 412 and quantum state 421 of energy storage system 420 may not be in an entangled state as a result of tuning 503, if the parity of multiple data qubits 411 had a single value prior to the procedure. Thus, ancilla qubit 412 may be a pure quantum state (either a |0) or |1) state), rather than being in a superposition state. Similar, quantum state 421 may be in a full number state (e.g., n quanta) or a reduced full number state (e.g., n - 1 quanta). Further, due to the relationship between ancilla qubit state and the parity of multiple data qubits 411, the pure quantum state of ancilla qubit 412 may indicate the parity of multiple data qubits 411.

[0105] Method 500 comprises measuring 504 ancilla qubit 412 to determine the ancilla qubit state. The ancilla qubit state reflects the parity of multiple data qubits 411. Due to the relationship between ancilla qubit state and the parity of multiple data qubits 411, the measurement of ancilla qubit 412 may indicate the parity of multiple data qubits 411 (or part thereof), or collapse them to a specific parity value. For example, the measurement of ancilla qubit 412 may indicate the ZZ parity of a subset of multiple data qubits 411 (i.e., two qubit of multiple data qubits 411). However, it is noted that the measurement of ancilla qubit 412 may correlate to other parity measurements. As an example, due to the relationship between ancilla qubit state and the parity of multiple data qubits 411, if ancilla qubit 412 is measured in the 0 state, this may indicate that the ZZ parity is even (indicating that the combined state of two qubits of multiple data qubits 411 is 00 or 11). Likewise, if ancilla qubit 412 is measured in the 1 state, this may indicate that the ZZ parity is odd (indicating that the combined state of two qubits of multiple data qubits 411 is 01 or 10).

[0106] In some embodiments, tuning ancilla qubit 412 establishes disentanglement between ancilla qubit 412 (e.g., the ancilla quantum state) and the changed quantum state of energy storage system 420 resulting in a pure quantum state of ancilla qubit 412 that reflects the parityof multiple data qubits 411. Disentanglement may be established by performing an energy transfer gate (i.e., a TT / 2 gate) on ancilla qubit 412.

[0107] In some embodiments, method 500 may further comprise, after tuning ancilla qubit 412 to disentangle ancilla qubit 412, further tuning multiple data qubits 411 to restore an original quantum state of multiple data qubits 411. The “original quantum state” of multiple data qubits 411 may refer to the quantum state that multiple data qubits 411 were in after tuning 502 of multiple data qubits 411. For example, further tuning multiple data qubits 411 may create entanglement between multiple data qubits 411 similar to tuning 502 of multiple data qubits 411. Further tuning may comprise perform a similar operation (e.g., apply similar gates) as tuning 502 of multiple data qubits 411.

[0108] Fig. 6 illustrates an example circuit (denoted as circuit 600) for determining an ancilla qubit state (of ancilla qubit 612) in relation to multiple data qubits 611. As can be seen in Fig. 6, energy transfer gate 631 may be applied to ancilla qubit 612 to entangle ancilla qubit 612 with the quantum state of an energy storage system. Energy transfer gate 631 may be applied by setting a detuning frequency of ancilla qubit 612 to about zero. Entangling gate 632 is then performed by tuning multiple data qubits 611 to transfer energy between multiple data qubits 611 to entangle multiple data qubits 611. Since entangling gate 632 depends on the quantum state of the energy storage system, the energy storage system (and thus ancilla qubit 612) receives a phase kickback based on the ZZ parity of multiple data qubits 611. Energy transfer gate 633 may then be applied to ancilla qubit 612 to rotate ancilla qubit 612 to a measurable basis, thereby establishing disentanglement between ancilla qubit 412 and the change quantum state of energy storage system 420. Although not shown in Fig. 6, ancilla qubit 612 may then be measured to determine the ancilla qubit state, which may be used to determine the ZZ parity of multiple data qubits 611. Circuit 600 is an example of a probing operation. More specifically, circuit 600 is an example of a probing operation of a ZZ parity.

[0109] In some embodiments, tuning ancilla qubit 412 may be performed to produce a superposition state of pure quantum states of ancilla qubit 412. Further tuning of ancilla qubit 412 may then be performed to produce the pure quantum state of ancilla qubit 412 from the superposition state. For example, as will be discussed in more detail later in the disclosure, tuning of ancilla qubit 412 may map the parity of multiple data qubits 411 onto ancilla qubit 412, producing a superposition state of | ^)even11) + 1 ^}odd10, where | ' / / ^cvcnand | ' / / )oddare the evenand odd components of the quantum state of multiple data qubits 411 and 10 and |1) are thepure quantum states of ancilla qubit 412. As such, the tunning of ancilla qubit 412 creates a phase kickback on ancilla qubit 412 depending on the parity of multiple data qubits 411. Further tuning of ancilla qubit 412 (such as a subsequent energy transfer gate) may establish a direct mapping between the parity of multiple data qubits 411 and the ancilla qubit state. Hence, measurement of the ancilla qubit state (i.e., the pure quantum state of ancilla qubit 412) may be used to determine the parity of multiple data qubits 411.

[0110] Understanding the parity of multiple data qubits 411 may be useful for quantum error correction. For example, the parity may be used to determine whether an error (such as a bit flip or phase change error) may have occurred one or more of multiple data qubits 411 has occurred during a quantum computation. Different parity measurements (ZZ, XX, XZ, ZX parity for two qubits, for example) may be used to determine the specific error (e.g., bit flip error or phase flip error) and which qubit the error has occurred. Therefore, in some embodiments, method 500 further comprises performing quantum error correction based on the determined ancilla state. For example, the determined ancilla state may indicate the parity of multiple data qubits 411 which may indicate a bit flip error on one of multiple data qubits 411. Repeating method 500 may determine another parity of multiple data qubits 411, which may be used in combination to determine which qubit has experienced the error.

[0111] It is noted that method 500 may be referred to as a probing operation. In a sense, method 500 may be used to probe any collective gate (essentially, by using a single collective gate). This property is particularly attractive for quantum error correction applications, which may use frequent probing of multi-qubit parity operators (stabilizers). As such, and as will be further discussed, method 500 may be used to “probe” stabilizers (i.e., stabilizing operations or multi-qubit parities) to perform quantum error correction. For example, the stabilizers may be related to a parity of multiple data qubits 411 and hence, performing method 500 may “probe” the stabilizer to determine the related parity by determining the ancilla qubit state.A method for creating a logical qubit

[0112] Fig. 7 illustrates an example method (denoted as method 700) of creating (or forming, defining or encoding) a logical qubit from multiple physical qubits on a quantum computing system comprising an energy storage system. Method 700 will be discussed with reference toFig. 4. It is noted that some embodiments of quantum computing system 100 described above may equally be embodiments of method 700.

[0113] A logical qubit is designed to be more robust against quantum errors than a physical qubit. In some examples, the logical qubit is in a Greenberger-Home-Zeilinger (GHZ) state. A GHZ state is a type of entangled quantum state that involves three or more qubits, which is useful in quantum error correction. The logical qubit may be encoded into multiple data qubits 411.

[0114] Method 700 comprises tuning 701 a resonant frequency of multiple data qubits 411 of quantum processor 410 using energy storage system 420. It is noted that tuning 701 the resonant frequency may comprise tuning a subset of multiple data qubits 411. Tuning 701 may occur in a similar manner as previously described. Tuning 701 multiple data qubits 411 then entangles 702 multiple data qubits 411. In other words, tuning 701 multiple data qubits 411 creates entanglement between multiple data qubits 411. For example, by setting the detuning frequency for each of multiple data qubits 411 to a similar (detuning) value, multiple data qubits 411 may become entangled, as previously discussed. In a sense, tuning 701 multiple data qubits 411 may cause multiple data qubits 411 to exchange energy amongst each other (e.g., in the form for quanta). Entangling 702 multiple data qubits 411 thereby creates 703 the QEC logical qubit. Creating 703 the QEC logical qubit may refer to encoding a logical qubit from multiple data qubits 411, in the sense that multiple data qubits 411 now collectively encode the quantum information of a single qubit with robustness to errors. In some embodiments, the logical qubit is in a ground state (e.g., a logical ground state).

[0115] Quantum error correction may be performed on the logical qubit using stabilizers (which may also be referred to as stabilizing operations). A stabilizer may be a collective operator of I, X and Z gates, and may be related to the parity of multiple data qubits 411. For example, for 4 data qubits, the quantum error correction [4,2,1] surface code stabilizer generators may be XXXX, ZZII and IIZZ. The stabilizer XXXX bound the X-parity of the 4 qubits, while the ZZII stabilizer bound the Z-parity of the first two qubits. These stabilizers may be “probed” using method 500 as discussed above by using ancilla qubit 412. As such, a parity of multiple data qubits 411 may be determined by probing the associated stabilizer. Therefore, method 700 may further comprise probing a stabilizer operation using a controlled-parity operation; and performing error correction of the logical qubit based on a result of probing the stabilizer operator.

[0116] For example, to determine the ZZ parity of qubit 3 and 4 of a 4-qubit logical qubit, method 500 may be performed using the IIZZ stabilizer. As such, the measurement of ancilla qubit 412 by performing method 500 may be used to determine the ZZ parity of qubits 3 and 4. The use of multiple stabilizers and probing these multiple stabilizers using method 500 may be used to identify whether an error has occurred and the qubit where the error has occurred.Quantum error correction may then be performed by repeatedly probing the logical qubit’s stabilizers.

[0117] In some embodiments, other operations may be performed to create (i.e., encode) 703 the logic qubit. For example, one or more collective gates may be applied to multiple data qubits 411 to create 403 the logical qubit. These collective gates may include a collective energy transfer gate or relative phase gate. These gates may be useful to ensure the logical qubit is efficiently created using energy storage system 420 (e.g., to transform the states into a useful form to create 703 the logical qubit). These collective gates may be performing in multiple detuning steps, as previously discussed.

[0118] Fig. 8 illustrates an example circuit (denoted as circuit 800) for creating (encoding) a logical qubit 840 on a quantum computing system comprising an energy storage system. As can be seen in Fig. 8, entangling gate 832 is applied to multiple data qubits 811 to entangle multiple data qubits 811 into logical qubit 840. Entangling gate 832 may be applied by setting a detuning frequency for each of multiple data qubits 811 to a similar value, as has been previously discussed. As can be seen in Fig. 8, circuit 800 comprises other gates (such as a collective energy transfer gate and a collective relative phase gate), which may be useful is the creation of logical qubit 840.

[0119] In some embodiments, the method further comprises performing a probing operation, wherein the logical qubit is in a superposition state after the probing operation. The probing operation may be similar to method 500. As such, in some embodiments, performing the probing operation comprises performing a controlled-parity operation of a subset of multiple data qubits 411 using ancilla qubit 412 of quantum processor 410. The superposition state may be a logical superposition state as it evolved from a logical state. The superposition state may be an X state, meaning that it is a superposition between the 0 and 1 state.

[0120] Fig. 9 illustrates an example circuit (denoted as circuit 900) for creating a superposition state of logical qubit 940 (e.g., a logical superposition state) on a quantum computing systemcomprising an energy storage system. Circuit 900 represents a probing operation of an IIXX stabilizer according to method 500. As can be seen in Fig. 9, a collective energy transfer gate is applied to two of the data qubits of logical qubit 940, as well as ancilla qubit 912. The collective energy transfer gate entangles ancilla qubit 912 with the quantum state of the energy storage system, while also transforming from the X to Z system for the two qubits of logical state 920, given that the natural gate of the system is Z related. The two qubits are then entangled by probing IIZZ stabilizer 932 (which may be considered to be a controlled parity operator), which create a relationship between the ancilla qubit state and the ZZ parity of the two qubits. The collective energy transfer gate 933 is then applied to transform the two qubits from the Z to X system, and rotating ancilla qubit 912 to the measurement basis. The state of ancilla qubit 912 is then measured using measurement 934, which causes the multiple data qubits forming logical qubit 940 to collapse into the logical superposition state.

[0121] Similar to performing error correction on the logical qubit (e.g., which may be in a logical 0 state), quantum error correction may be performed on the superposition state using stabilizers. Therefore, after creating the superposition state, method 700 may further comprise probing a stabilizer operation using a controlled-parity operation; and performing error correction of the logical qubit based on a result of probing the stabilizer operation. For example, the logical superposition state may be encoded into 4 qubits as discussed above. The stabilizers for the logical superposition state (i.e., the logical X state) may be XXXX, ZZII, IIXX and IIZZ. By using probing these stabilizers in combination, an error may be identified as well as the qubit where the error occurred.Example embodiment

[0122] An example embodiment of the disclosed quantum computing system will now be discussed. This example embodiment was used to determine the results discussed later in the disclosure. However, it is noted that the results presented herein generalise to other embodiments of the disclosed quantum computing system. In the example embodiment, energy storage system is a quantum battery. More specifically, the quantum state of the energy storage system is a bosonic mode in a Fock state, i.e., energy storage system is a shared bosonic mode coupled to qubits via the Tavis-Cummings Hamiltonian. The Tavis-Cummings Hamiltonian describes the interaction between an ensemble of identical two-level atoms and a single-mode quantised bosonic field, such as a photon field.

[0123] In the example embodiment, when the energy storage system is initialised in a Fock state, the unitary time-evolution of the system is constrained to an excitation-preserving subspace of the full Tavis-Cummings Hilbert space to align with the qubit space. Additionally, specific relative phases between qubit eigenstates are maintained, and three switchable gate types are enabled by tuning the qubit frequencies dynamically. Namely, the following gates are constructed: (i) energy -transfer gates for resonant energy exchange between qubits and the battery, (ii) all-to-all entangling gates in the dispersive regime, and (iii) relative phase or idling gates via large detuning.

[0124] Fig. 10 illustrates an example embodiment of the disclosed quantum computing system, (a) The quantum battery is represented as a bosonic mode in a Fock state, coupled to a set of qubits (each Bloch sphere represents a qubit). The state of the battery corresponds to a fully charged battery (the Fock state when all qubit state is |0)) minus the number of qubits which are in the 11) state, (b) When the battery is in a Fock state, the system’s total number of quanta, determined by the fully charged battery state, remains conserved throughout the computation. This conservation reduces the system’s Hilbert space dimension to 2N(where N is the number of qubits), mirroring the dimension of an N-qubit system, (c) Quantum computation is performed in a switching manner, controlled by the energy detuning (A) between the qubit’s energy (a)q) and the battery resonance (coj,). (d) In energy-transfer gates(u)q = cor), energy is exchanged between the battery and the qubits. Collective effects between the qubits enables superextensive energy transfer in certain cases, (e) In entangling gates, dispersive coupling facilitates multiqubit entanglement with all-to-all connectivity via the bosonic mode.Results and mathematical formulationThe Tavis-Cummings Hamiltonian for quantum computation

[0125] To investigate how a Fock-state quantum battery facilitates quantum computation and to translate the Hamiltonian into qubit gates, the Tavis-Cummings model is adapted by expressing it in terms of dressed mode-qubit operators. The system comprises a bosonic mode, serving as the quantum battery, with frequency coj, coupled to N two-level systems (qubits) with frequencies cOj. Under the rotating wave approximation, the system is described by the Tavis-Cummings Hamiltonian,NH = ha)bd^a +with and a being the bosonic mode’s creation and annihilation operators, respectively, and is the coupling constant between qubit 1 and the resonator. The operators a* = 11 t ) ( 0t1 and af = 10j)( | correspond to the raising and lowering operators of the ith qubit. The direct product of the qubit states and the boson number state |s) ® \nb) forms a basis of the quantum computation where nbrepresents the number of quanta in the quantum battery and |s) =... sn) are the spin quantum numbers of the qubits with E 0,1.

[0126] A unique feature of the system’s Hamiltonian is its conservation of the total excitation number operator. Because any creation (or annihilation) of a qubit excitation is accompanied by the annihilation (or creation) of quanta in the battery, [H, nex] = 0, where nex= a^a + <$t <$t counts the number of excitations in the combined quantum battery qubit system, which describes the “full-battery” excitation number. Consequently, the time-evolution of the initial state which is a qubit ground state and a full-battery Fock state, |0)®w0 lnex) is restricted to the 2Ndimensional Hilbert space which satisfies nex= nex. This result is directly derived from the fact that the creation (or annihilation) of a qubit excitation is accompanied by the annihilation (or creation) of quanta in the battery. As a result, the Hamiltonian may be block diagonal according to the nexvalue, as shown in Fig. 11. This subspace has dimension 2N, and its dynamics map may exactly onto the 2N-dimensional Hilbert space of N qubits, thereby enabling quantum computation with energy recycled entirely within the subspace. Although counter-rotating terms may couple between subspaces, their contribution is negligible and can be compensated when g, / ct)j" 1% for all qubits (derived in Appendix S6).

[0127] Fig. 11 shows the block structure of the computational subspaces. Choosing a single value for the total number of excitations in the joint energy storage system -multiple qubits system, nex, may be sufficient to reach a computational subspace, where energy is recycled throughout the computation. When nexis smaller than the number of qubits N, the quantum battery may lack sufficient energy to support full quantum computation. Transitions between blocks may be possible through energy loss, injection of external energy, or via counter-rotating terms.

[0128] To describe how the creation (or annihilation) of a qubit excitation accompanied by the annihilation (or creation) of quanta in the battery, the dressed operators are introduced oj£=and ®d,i=■> where o^ ==Enb\nbHnb + 1|- In Appendix Section SI, it is proved that and o^£satisfy the Pauli algebra only if the state |0£, Q battery) is excluded from the subsystem states, which is always satisfied when nex> N. Within this nex-subspace, the system Hamiltonian can be expressed exclusively in terms of the qubit dressed operators asNhgi (fd,i ^ex "r nex?lq&d,i I ■ (2)i=l

[0129] Here, A£= co£— (x>brepresents the frequency detuning between qubit i and the mode, nexis treated as a scalar due to its conservation during the computation, and the dressed qubit excitation operator is expressed as nq= ' t^d.i d.i- Inthis disclosure, computations are performed using Eq. (2), assuming gt= g for all qubits.

[0130] The interaction term in Eq. (2) encompasses the unique features of performing computation with quantum batteries. First, the initial state of the battery, as well as its state during the computation (nb= nex— nq), plays an integral role in the computation so that the gate duration would depend on the battery state. Second, the quantum battery enables non-local computation when coupled to all qubits. The inseparability between nqand oj£indicates that the state of all the qubits may determine which gate is implemented given a specific set of A£values and gate duration. However, it is shown below that high-fidelity computation is achievable solely through energy detuning. For energy transfer gates between the qubits and the battery, the detuning is set to approximately zero. Relative-phase gates are implemented by choosing slightly different detuning energies for each qubit, ensuring that all qubits are off-resonance from one another. Entangling between qubits is achieved when all qubits to be entangled are detuned to the same energy A.Superextensive gates

[0131] A notable characteristic of quantum batteries is their superextensive speed-up in energy transfer. These gates are analysed by setting A£= 0 for all qubits, reducing the Hamiltonian to fag l^=r(®dpjnex ~ nq + yjnex ~ ^q^d.t)- F°r asingle qubit, this Hamiltonian simplifies to hg^rex^p recovering the expected Jaynes-Cummings result: the battery can executes an Xgate when the interaction is applied for t = — y=, with the expected nexspeedup in the gate time. The superextensivity arise as qubits are added to the system, enabling collective gates that map between any two bright-states of the collective system, such as |0)®w«— > |1)®W. These gates show an additional superextensive speedup, as illustrated in Fig. 12a. The gate time decreases with an increasing number of qubits, assuming a fixed number of battery quanta per qubit (nj, / N). Increasing this number may reduce the gate time, eventually converging to the superextensive speed-up limit of 1 / V which correspond to a single-qubit gate duration of TC—=. Superradiance in optical cavities, demonstrated in systems like superconducting qubits provides a specific example of these collective gates. Quantum computation with quantum batteries suggests that superradiance and superabsorption can play an integral role in speeding up quantum computation.

[0132] Figs. 12a and 12b shows superextensive charging of the qubits. Fig. 12a shows the collective charging time versus the number of qubits for different ratios of battery quantum per qubit. Collective qubit charging (|0)®w-> |1)®W) time for different number of battery quanta per qubit (blue dots), normalized to the parallel charging time (pink dashed). Values below 1 indicate superextensive charging. When the battery is well-populated relative to the number of qubits (simulated here up to 9 quanta per qubit), the charging time approaches the superextensive limit of l / / (orange), demonstrating that increasing the number of qubits reduces the gate time. Fig. 12b shows the collective charging gate error versus battery quantum per qubit. The collective charging gate error (defined by the normalized energy transfer to the qubits) as a function of the initial number of quanta in the battery. Results are shown for the battery initialized in a Fock state (blue shades) and in a coherent state (green shades). The Fock-state quantum battery exhibits significantly better performance than the coherent- state quantum battery, with gate error decreasing further as more qubits are added to the system. In both (a) and (b), all qubits are on-resonance with the battery.

[0133] A surprising effect of these collective gates is that increasing the number of qubits also improves gate fidelity, as shown in Fig. 12b. When nex» N, the Hamiltonian in the zeroth-order approximation becomes separable and performs a perfect collective X-gate. Indeed, it is found that the HJ, / N ratio is the primary parameter in reducing the collective gate error, which achieves practical error values between 10-2and below 10-3. Such values are unattainable when the quantum battery is initialized in a coherent state, where the nexvalues are coherentlydistributed in the initial state (as indicated by the green shades in Fig. 12b). This highlights the advantages of the quantum battery to be initialized in a quantum state of light overcoming previous bounds on the gate fidelities and minimal energy requirements which assumed an external coherent energy source.

[0134] Notably, a perfect collective charging gate (a |0)®w«— > |1)®Wmapping) is always achievable when using a Fock-state battery and a single qubit with a single detuning step, as well as a perfect charging of N qubits which are charged in N detuning steps. The example in Figs. 12a and 12b demonstrates a specific state mapping that pass through Dicke symmetric states (bright states). In the Appendix section S2, a separable XNgate may be applied when nex» N 2with an average gate error per qubit is approximately 2 (-) —. Although a large-nj, battery may \8 / 71^be useful, such a circuit reaches low-error unitary computation even with an nb / N ratio of below 2 when optimizing the detuning energies over time. Using a small Fock number is useful to reduce the battery lifetime which scale as l / nb.Entangling and control-parity gates

[0135] Entangling multiple qubits in the dispersive regime has been demonstrated across various platforms with similar settings. This entanglement gate is performed when the qubits are detuned from the mode’s resonance by a similar frequency A. The entangling implement gate may be implemented as either an iswap gate or an ^iswap gate when the gate is applied for t = 2 -2.]^ and t =4respectively, assuming the bosonic cavity is initially empty. However, asdiscussed below, the quantum battery’s initial state can modify the executed entangling gate.

[0136] By converting Eq. (2) to angular momentum operators, and implementing the Schrieffer-Wolff Transformation when A » g^ nex(see Appendix section S4), the dispersive Hamiltonian of the system becomes,where ]z==7—and / ±= Eq- (3) includes the exchange term q2 A(— / - / +) between qubit states with similar nq(number of qubit excitation). A full exchange ofquanta between these states occurs when the interaction is applied for tdisp=2 2 / A’regardless of nex. The quantum battery state and the detuning value influence the exact gate executed through the relative phase for different Jzvalues. These relative phases exhibit jumps of - when A2the ratio — is an integer (as was shown in ion-based systems) jumps of n according to nex. For example, for two qubits, the entangling gate is(— l)ne* 0 0 0 ^dispC^-ex’ A) — exp( iffdispfdisp)—(4) 0 -i^2 / a20 0which equals an iswap gate only when A2 / g2mod 4 = 3 and nexis even.

[0137] The dependence of the executed entangling gate on the quantum battery’s initial state is useful for probing a multi-qubit parity with a single entangling gate. The procedure is explained in Fig. 13. In the first stage of the protocol, an ancillary probing qubit is entangled with the battery using a tt / 2 energy -transfer gate, creating an entanglement between the ancillary qubit and the battery state. Subsequently, the entangling gate becomes a controlled-parity unitary gate when applied for t = tent, where the controlling qubit is the originally ancillary qubit (which is derived in Appendix section S5):u = |t / disp(nex) (|0a)(0a| 0 I + I la)(laI 0 ([ / disp(nex))’1^disp(nex - 1) ), (5)where (t / disp(nex)} ^disp(nex—1) = e21tdlsp~J = iNz®Nwith N being the number ofqubits in the entangling gate. When applying another TT / 2 energy-transfer gate between the ancillary qubit and the battery, a direct mapping is established between the parity of the N qubits and the ancillary qubit state (i.e., a phase kickback). Therefore, the system can probe any Z®woperator (for any N) with a single collective gate, independent of the cavity's state.

[0138] Fig. 13 illustrates an 0(1) multi-qubit parity probing protocol according to the present disclosure, (a) Energy-level illustration of the protocol for probing the ZZ parity of qubits Qi and q2using q3. In the first step, q3is entangled to the battery which in turn reach superposition of n and n — 1 quanta. Next, q3is highly detuned from the cavity while Qi and q2perform anentangling gate. Since this gate depends on the battery state, the battery (and thus q3) receives a phase kickback based on the ZZ parity of Qi and q2. Finally, another energy -transfer gate between the battery and q3correlates q3’s state and the ZZ parity, (b) Quantum circuit corresponding to the protocol illustrated in (a). The bottom panel highlights the role of the quantum battery state in the computation, (c) The number of required gates per stabilizer scales as 0(1), regardless the stabilizer weight (the number of qubits involved in the stabilizer).

[0139] Fig. 14 demonstrates a specific circuit (a) and its simulated implementation (b) when probing a two-qubit parity. In this simulation, an 10 / ®H gate is chosen to create an entanglement between the ancillary qubit ( q~, ) and the energy storage system and then to disentangle them after the controlled-parity gate. The dispersive gate implements CZ and SWAP gates, which are accompanied by a set of local Z gates if qubit c / 2is in the 1 state. Eight detuning steps were used to implement the I I H gate with an overall duration of n 14g (inset of Fig. 14b). In the dispersive gate (and the restoring step),01were detuned toA = 27.5g for a duration of 27.5;r / 2g, while c / 2was heavily detuned to minimize its effect. Comparing the simulated circuit with an ideal parity -probing circuit (Fig. 14a), a Hilbert-Schmidt distance of 0.642 is obtained, corresponding to an average circuit fidelity of 95.4%. Fidelity may be further improved by allowing continuous time evolution, which adds degrees of freedom.

[0140] Fig. 14 illustrates multi-qubit parity probing through battery-dependent dispersive gates. (a) The protocol for probing multi -qubit parities through dispersive interactions, exemplified with a 2-qubit probing circuit. Step (1) entangles the ancillary qubit (< / 2) with the battery. Step (2) applies a battery-dependent dispersive gate that implements a superposition of ^disp ^ex)and ^disP(wex “ inthe joint battery-two-qubit subspace. Step (3) maps the parity of onto the ancillary qubit, producing kLn I ^2+kL I °)?2’whereLnandkLare theeven and odd components of, respectively. Step (4) restores the probed qubits to their original state |i / / ^. This sequence remains unchanged for any 2-qubit parity probing, (b) Implementation of the circuit in (a) using only qubit frequency detunings, for nex= 5. The A 7Tduration of each dispersive gate is - (with A = 27.5g) independent of the number of qubits g ^gprobed. The I ® I® H gate takes approximately -, implemented here in 8 steps (inset of 22gFig. 14b).Completing a universal quantum gate-set

[0141] As previously discussed, local unitary gates become particularly challenging in a collective system, where the state of the quantum battery — and thus the state of all qubits — can influence the unitary transformation achieved by a given single-qubit detuning for each duration. Thus, the goal of the disclosed quantum computing system is to reach a similar battery-qubit energy exchange gate when the subsystem includes nexquanta, down to nex— N + 1 quanta (when the battery is initialized in the Fock state).

[0142] In Appendix Section S8 and Fig. 20, it is demonstrated that few detuning steps may be used to implement a local non-trivial energy-changing gate that produces the same unitary regardless of the states of the other qubits. In experimentation disclosed herein (including examples with up to five qubits and a quantum battery fully charged with seven photons) two detuning steps were sufficient to achieve average gate fidelities of above 99.8% and worst-case fidelities of 99.5%. Incorporating additional detuning steps increases the available degrees of freedom, which can be exploited to further enhance fidelities and accommodate larger systems. From a theoretical standpoint, the quantum battery system can implement any N-qubit target 4W-1unitary using M > — — detuning steps since the total degrees of freedom, (N + 1) X M (N qubit detuning values plus the step duration), accede the unitary degrees of freedom. This approach may be used for a complex unitary, though the exponential scaling in the number of qubits is greatly mitigated for the purpose of showing a universal gate set. For example, the simulations disclosed herein for a five-qubit system used only two detuning steps rather than the 170 suggested by the theoretical bound.Simulating a quantum error correction circuit

[0143] To demonstrate the implementation of quantum computation with quantum batteries using the native gates described above, a 5-qubit system that encodes the logical states of the d = 2 quantum error correction (QEC) surface code is simulated (see Fig. 15). The system’s evolution is simulated according to Eq. (2) with parameters nex= 7 and g = 2n • 0.015 GHz (further simulation details provided in Appendix section S9). These parameters representsuperconducting systems. The quantum gates are implemented exclusively by adjusting the qubits’ detuning frequencies to an optimized A = (A1;. A5) with an optimized step duration. These values are chosen in most cases to set lower qubit frequencies compared to the resonator frequency to avoid the |1) -> |2) qubit transitions via the cavity. The optimisation achieves better performance compared to the results in Fig. 15 by enabling high-fidelity gates which incorporate energy-transfer, entangling, and relative phase when Aj~g. Additionally, the optimisation enables gates to be executed in multiple steps, enabling the realization of effective local gates even when using an inherently non-local Hamiltonian. Fig. 15b presents the detuning frequencies over time.

[0144] Fig. 15 illustrates encoding a QEC logical state, (a) Left: The distance-2 QEC surface code, defined by three stabilizers. Right: An example of a superconducting qubit configuration that could enable the computations, (b) The detuning energies of the five qubits over time. The procedure first encodes the 4-qubit GHZ state, representing the logical |0) state of the QEC code. Then, by probing the 11XX stabilizer with qubit q5, the state then becomes the logical |+) state, (c-d) Stabilizer values and logical state fidelities after encoding the logical |0) (c) and |+) (d) states, (e) The exact quantum circuit implemented in (b), where UentN = 4) denotes the 4-qubit entangling gate. Note that the measurement of q5is not shown in (b) but is required to collapse Qi—Q4 intothe logical |+) state. Additionally, some energy -transfer gates consist of multiple steps due to the inherent Hamiltonian non-locality.

[0145] The simulated QEC code includes four qubits, where the code space is defined by three stabilizers, XXXX, ZZII, IIZZ), for which the qubits are eigenstates with eigenvalue 1. Encoding the logical state into |0) or |+) requires the qubits’ state to also be an eigenstate of a fourth stabilizer, 1ZZ1 or 11XX, respectively. Since the logical 10) is a 4-qubit GHZ state, it is encoded using a single entangling gate, which a fidelity of 0.998 is achieved. Next, an ancillary qubit is utilised and the controlled-parity procedure to probe the 11XX parity. This operation collapses the state into a ±11XX eigenstate with eigenvalue 1 upon measuring the ancillary qubit, where the sign is determined by the measurement result. Notably, this mid-circuit measurement is crucial as it modifies nexin real time, directly influencing the implementation of the following gates. Overall, the circuit successfully encode the logical |+) state of the code using only two entangling gates, achieving a fidelity of 0.981. The quantum circuit which is implemented by the qubit frequency tuning is shown in Fig. 15, where t disp(N = 4) denotes the 4-qubit entangling gate, and the controlled-parity gate is depicted with dashed lines.

[0146] This simulation demonstrates the potential of quantum batteries in facilitating QEC algorithms, enabling computation using only flux and readout control, while leveraging all-to-all connectivity. Such connectivity is particularly appealing for low-density parity check (LDPC) codes, which may offer improvements in the logical-to-physical qubit ratio. Additionally, the system's native iswap entangling gate was found to relax hardware requirements for surface codes. However, the results also highlight challenges in scaling this concept. First, achieving superextensive speedup may be achieved using a large nex, for which a large detuning value A »> 7nex may be to remain in the dispersive regime. This, in turn, increases the timescale for entangling gates which are linear with A. Second, increasing the number of qubits per battery creates a challenge in implementing local gates, even with several steps.Around-Resonance Multi-Qubit Gates

[0147] Previously in this disclosure, it has been shown how the entanglement between a populated energy storage system and the qubits can be utilized for unique processing. However, the timescales of the dispersive gates might introduce errors in their experimental implementation, especially when increasing the number of qubits. This effect motivated efforts to find new computation methods for shared-resonator quantum computers using driven cavities. The energy storage system may facilitate computation without reaching the dispersive regime timescales, but rather through energy exchange processes between the energy storage system and the qubits, reaching overall unitary durations of a few n / g.

[0148] Fig. 16 illustrates computation via battery-mediated multi-qubit interactions near resonance. Fig. 16a presents a circuit example which involves energy transfer from the energy storage system to the qubits and a non-Clifford three-qubit entangling gate in the form of a Toffoli gate. In conventional quantum processors, this circuit is broken down to the hardware’s native gates as shown in Fig. 16b. A similar approach may be used to implement the circuit using an interleaved sequence of energy -transfer gates and entangling CZ gates, shown in Fig. 16c. Each energy -transfer gate is implemented in three O.lir / g steps. The CZ gates were implemented through five steps of n 12g, intuitively to enable a back-and-forth interaction between two qubits via the energy storage system. Overall, this simulation implemented the circuit with an overall infidelity of 3 x I O ', where the detuning parameters were chosen to optimize the fidelity.

[0149] Fig. 16e shows that the overall circuit time can be further reduced and the implementation can be simplified when combining the whole circuit into a single SU(8) unitary (Fig. 16d) implemented in few detuning steps. Specifically, six nl 4g steps are used to reach an infidelity of 3x 109with an overall duration of 4.5x! g. This example shows that the shared-resonator can combine multi -qubit entangling and energy transfer from the energy storage system to the qubits within the same sequence. This property can open paths to new computation methods with smaller circuit depth that utilize the all-to-all connectivity of the shared-resonator architecture, as opposed to a shared-resonator pairwise computation.

[0150] Fig. 16 illustrates computation via battery-mediated multi-qubit interactions near resonance, (a) Target circuit featuring a three-qubit Toffoli gate, applied after preparing the control qubits in superposition, (b) The standard decomposition of the target circuit into a sequence of single-qubit and pairwise entangling gates, (c) Implementation of (b) yielding a final infidelity of 3 x I O '. Energy-changing operations (thin arrows) are realized in three detuning steps, each with a duration of 7r / 10g, while entangling operations (thick arrows) require five steps, each of duration x! g, to account for round-trip energy exchange via the battery, (d) Uniting the full target circuit into a single multi -qubit unitary, (e) Implementation of (d) using six steps of duration 3^- / 4g, reaching an infidelity of 3x 109while significantly reducing the total circuit time compared to (c).Experimental Feasibility with Superconducting Qubits

[0151] This section discusses the feasibility of experimentally realizing the disclosed system, focusing on an implementation with superconducting transmon qubits.

[0152] To assess resilience against system noise, the detuning sequence from Fig. 16e was stimulated under realistic noisy conditions, and achieved a final state fidelity above 99.2%. The system’s evolution was modelled with typical superconducting parameters of g = 15 MHz, corresponding to a total evolution time of 150 ns (six 25 -ns steps), a cavity lifetime of 600 p s, single-qubit 7] and 7)* times of 30 p s and 10 / s, respectively, and a 5-photon Fock state purity of 99%. The simulated fidelity remains unchanged even when including additional phase drifts with a standard deviation of 0.005 rad / ns. These results suggest that superconducting hardware has reached a maturity level sufficient, in principle, to demonstrate quantum information processing with energy storage systems.

[0153] High-fidelity initialization of the energy storage system in a Fock state may be achieved with several existing protocols. Any charging protocol requires a nonlinear element, which can be implemented using one of the qubits coupled to the energy storage system and its drive line acting as the drive to the energy storage system. Maintaining a small Fock number is useful, as the energy storage system lifetime scales as 1 / nb. This is especially useful in this disclosure as any energy storage system decoherence may be converted to qubit decoherence due to their entanglement during the computation. The circuit examples used in this disclosure had a ratio of nb / N < 2 to achieve low-error unitary operations by optimizing detuning values over multiple steps. Moreover, since the disclosed system operates on timescales of order I g, the additional errors due to photon loss remain manageable. Executing computations may utilise calibration of the optimized flux parameters for each device, due to fabrication variances. In particular, flux profiles may be avoid ringing or overshoot by smoothing the signals during rise and fall times, which can be in the order of 5 ns to be compatible to the control electronics and the typical timescales of the disclosed system.

[0154] The parameters of Fig. 16e were also used to evaluate the contribution of counterrotating terms beyond the assumed Tavis-Cummings Hamiltonian (see Appendix Section S6). The procedure for quantum computation remains quantitatively accurate as long as the coupling ratio satisfies g I ab" 1%. As shown in Fig. 18, recalibrating the step durations by 0.5% per step can reduce errors from counter-rotating terms by nearly two orders of magnitude. Their small effect arises because the counter-rotating terms do not create energy-transfer between blocks as at least: 100 counter-rotating oscillations average out during the: 7i I g detuning step duration.Scaling opportunities and energetic efficiency

[0155] Quantum batteries offer a unique opportunity to scale quantum computing with superconducting qubits by enabling full qubit control through flux lines and supplying gate energy prior to computation. Fig. 17 illustrates the notable differences between conventional and quantum battery architectures: in the quantum battery approach, all qubits are connected to a shared superconducting cavity, eliminating the need for individual qubit drive lines and their associated attenuators which generate active and passive heat during conventional computation. This reduction in heat sources increases the potential number of qubits per cryogenic fridge, addressing a notable bottleneck in scaling cryogenic quantum computation.

[0156] To analyse the impact of flux-only control, the heat power consumption of the lowest two layers in a cryogenic fridge during quantum computation is evaluated (details provided in Appendix section S7). The active and passive heat generated by the control cables and attenuators was evaluated for different configurations of drive, flux, and readout lines, using standard pulse usage, see Fig. 17 (further detailed in Fig. 19). By comparing the total heat power to the cooling power of cryogenic fridges, it was determined that the potential scaling benefits of the shared-cavity architecture, as shown in Fig. 17. This analysis revealed that with current cable technology, the shared-cavity architecture could increase by 20% the number of qubits per fridge, with the primary heat source in the coldest cryogenic stage being attenuation along the flux lines. When superconducting cables are used for flux lines — which are unsuitable for drive lines due to the need for attenuators — the scaling factor increases significantly, reaching X8.75 compared to the standard configuration and X4 compared to the standard configuration with superconducting cables. In the case of a shared-cavity with superconducting cables, the heat consumption becomes dominated by the readout cables, including resonator drives and amplifier pumps where the rest of the heat is due to the passive heat within the drive line per battery (assuming 10 qubits per battery).

[0157] Fig. 17 shows the heat analysis and scaling opportunities of the shared-cavity quantum computation, (a) The shared-cavity (top) and the standard nearest-neighbour (bottom) architectures. Other computation includes an additional control line per qubit (drive) and corresponding attenuators which create passive and active heat during the computation, (b) Distribution of heat power at the cold-plate cryogenic level, determining the maximum number of qubits it can support. The pie chart’s area represents the total heat power, (c) The maximum number of qubits that a cryogenic fridge can support, derived from (b). The shared-cavity configuration enabled a factor of 1.23 additional qubits with available cables, while superconducting flux lines may enable increasing the number of qubits by a factor of 8.75 (or 4) compared to the standard configuration (with superconducting cables), (d) The accumulated room-temperature (RT) energy per qubit that is required for the input control lines as a function of the circuit depth. Although the quantum battery requires charging prior to the computation, removing the drive control keep the additional energy dominated by readout and the shared-cavity computation becomes energetically efficient after few QEC cycles. Detailed derivation of (b-d) is provided in Appendix section S7.

[0158] In addition to enabling quantum computer scaling, quantum batteries offer opportunities to reduce the total energy consumption of quantum computation. In this analysis, it wascalculated that the power consumption within the cryogenic fridge to estimate the computation's input energy at room temperature (detailed in Appendix section S7). The results, presented in Fig. 17, show that the shared-cavity architecture becomes more energy-efficient than other architectures once the circuit depth reaches approximately 10 QEC cycles. The shared-cavity architecture incurs a fixed overhead energy cost for initializing the cavity in a Fock state prior to computation. However, during computation, instead of the energy consumption which is sourced by both readout (measurement) and drive pulses, the quantum battery architecture exemplifies the concept of quantum field recycling for single-gate rotations, remaining solely with readout pulses. Notably, the energy for these pulses cannot sourced in the quantum battery since the measurement operation is not unitary. Nevertheless, the accumulated energy per circuit depth for the quantum battery architecture is significantly lower, and the difference in energy consumption between the two architectures becomes increasingly pronounced as quantum computation scales and circuit depth grows. In a more fundamental level, since the mid-circuit readout is used to stabilize the quantum system in the QEC procedure, the only energy that enters the quantum system during the computation with the quantum battery is purely to remove entropy from the system.

[0159] The heat and energy results described herein can be directly incorporated into previous holistic derivations of the energy cost of quantum computation. While some quantum computers are dominated by cryogenic cooling rather than active computation, the balance between cooling and computation depends strongly on system architecture. The large drive channel-to-qubit multiplexing of the energy storage system architecture meets the high-level design of the expected energetically-efficient quantum computation. In particular, the qubit-enhancement ratios described herein for a fixed cooling power can equivalently be interpreted as reduced cooling demand for a given quantum algorithm. From an algorithmic perspective, the disclosed system decreases the average power per qubit, which may be further reduced through circuitlevel optimisation.Summary and discussion

[0160] This disclosure details how quantum batteries can be utilized for quantum computation. A concrete example demonstrating fundamental concepts of quantum batteries is provided, including gate reversibility, the repeated involvement of the battery in unitary gates, and superextensive collective effects. These superextensive effects highlight the advantage of the Fock state over the coherent state in terms of gate fidelity, achieved through the in-phaseevolution of all qubit-battery dressed states. This coherence enabled the native entangling and control-parity collective (nonlocal) quantum gate set in the system. Using these gates, a QEC circuit is simulated with an entangling gate per multi-qubit parity measurement, employing only the qubits' frequency control. As a result, a scaling potential of *4 in the number of qubits per fridge demonstrates and it is shown that accumulated energy consumption during computation is dominated by readout, which may be used for qubit stabilization. Achieving heat generation only during readout offers a pathway to reaching the thermodynamic limit for energy -efficient quantum computation.

[0161] While this analysis primarily focuses on flux-tuneable superconducting qubits, the approach may also be applicable to other platforms, such as fixed-flux qubits when modulating the coupler or semiconductor spin qubits with energy tuning via external fields or atoms and ions within cavities where the cavity frequency is tuned. The energy -tuneable computation offers additional practical advantages for cryogenic quantum systems by eliminating drive lines. Such lines scale linearly at two lines per qubit at room temperature, leading to cumbersome setups and significant signal crosstalk. Furthermore, classical information required to optimize analogue drive pulses — such as those involving phase modulation — is entirely eliminated. The analogue precision required for drive lines makes control systems complex, expensive, and limited to room temperature. By relying solely on flux control lines, digital control becomes feasible, enabling full quantum control through cryo-CMOS technology and further reducing energy consumption. Finally, alternative readout techniques, such as microwave photomultipliers could further minimize heat and energy consumption. Altogether, quantum batteries represent a revolutionary architecture for cryogenic quantum computers, offering energetically favourable, simpler, and more scalable quantum computation.AppendicesSI. The Tavis-Cummings Hamiltonian with dressed operators

[0162] In this section a 2NX 2NHamiltonian is derived which describes an interaction between N qubits and a bosonic mode which acts as a quantum battery which facilitates all unitary gates in the qubit system. Assuming a system which includes a resonator with frequency coj, coupled to N 2-level systems (qubits), where each qubit has a tenable frequency ooj as flux-tunable transmons. The Hamiltonian that describes this system under the rotation wave approximation is Tavis-Cummings Hamiltonian (taking h = 1):H = a)ba^d + (SI)with and a being the resonator photon field creation and annihilation operators respectively, gt is the coupling constant between qubit 1 and the resonator, < J+ = |l£)(0£|, and a = |0£)(l£|. A general state in this system includes the qubit systems state and the cavity state, described as \s,npfl) where |s) = |s0Si -sn) with ste {0,1}.

[0163] A derivation is now provided that re-writes the Hamiltonian in a form which makes the collective qubit effects more visible and focuses on the qubit system. The following operators are defined: nb= c a as the photon number operator, nq=as the qubit excitation counting operator, and nex= nb+ nq. In addition, the operator ob= nblnb+ I is defined so thata+|nfc) = y / nb+ l\nb+ 1) = jfi^\nb+ 1) = y / fi^b\nb)where the theorem that if |r|r) (in this case \nb+ 1) is an eigenstate of 0 (nb) with eigenvalue A(nj, + 1) is used, then |I]J) is also an eigenstate of f(O) (in this casewith eigenvalue f(A) ( / nb + 1) when f(x) is smooth. This result leads to the equalitiesa+a£\s,nb) = \s,nb) = nex- nqd^di\s,nb),and similarly, when applying a complex-conjugate,<j i a = <j i obW-ex ^lqwhere ab= (a£)t= + 1| and (^ex ~=yj^ex ~ nqdue to the fact that hex— nqis Hermitian.

[0164] With these equalities, the system’s Hamiltonian can be written in a form which directly shows that all qubits interact with each other in a non-linear manner through the cavity,N 'N H = a>bd^d + > a>La^d- h-ex riqO'btit (S2)

[0165] It can be shown that this all-to-all interaction creates beneficial collective gates such as collective X gates and collective entangling gates. To further simplify the Hamiltonian, the following substitutions are made: d^d = nex— nqand oo£= a>b— A£, where A£is the qubit’s detuning from the resonator,

[0166] The first assumption in the derivation, which the following equations rely on, is that the initial state of the system is |0)®w0nex)-. corresponding to the ground state of the qubit system and a Fock-state with nexexcitations in the resonator. This assumption converts the cavity from being in a general state with distribution of values into a quantum battery. Moreover, when the eigenvalue of nexshould have a single value in the initial state, the system should remain with this eigenvalue throughout the computation since [H, nex= 0. Keeping the calculation within a single block of the Hamiltonian creates a direct mapping between the qubits’ subsystem state and the number of photons in the cavity (for a two-qubit system, the states are |00> ®|n>, |01> ®|n — l), |10) ®|n - 1>, |11> ®|n - 2).

[0167] In this scenario, nex= n is considered as a scalar throughout the computation, leadingNHn= u)fcn - + 71 Tlq(5b(5i j,i=land since u)j,n is a global phase to all states within the subsystem, the corresponding Hamiltonian can be written asN71 tlq ob(S4)i=l i=l

[0168] Finally, this Hamiltonian can be written with dressed operators which should obey Pauli algebra,°d,i = °b °t ■ °d,i = °b °ir(s5)

[0169] Noticing thatb i ' = hwhich leads to®b ®b ®b ®b ^d,i^d,i>so that the first term of the Hamiltonian and nqcan be written with cr* 0. The commutation relations obey[<5d,i,°d,i] = = [#ph#+>#ph#r] = [vph^phjvr^ + ^ph^^ri = | O 0ph)(0i' Opftl + [ri£+, < J£“].

[0170] Therefore, the operators oj£and 87 i could keep a Pauli algebra only if the state |0£, 0bis not part of the subsystem states. This condition is reached by assuming that n > N (the eigenvalue of nexis larger or equal the number of qubits in the system) so that 0 cavity photons are reached only if the qubit state is 11)®Nand n = N. Then, [< J^£, ad i\ = [< J£+, < J£“ ]. This assumption is also required to reach all qubit states during the computation.

[0171] These commutation relations enable one to define the Pauli operators= °d,t + °a,t = (°d,i°2,d; = 9d,£riJ£rid,£rid,£which satisfy all Pauli-operator identities and commutation relations. Finally, the Hamiltonian of the subsystem may be written as¥dAi + / 9i pdj n - nq+ n - nqad i). (S6) i=l ' N V '1=1

[0172] This equation is provided as Eq. (2). Computation in this disclosure are done with Eq. (S6) with 3i = g for all qubits. For all energy transfer gates between the qubits and the environment, the detuning is taken to ~0 for all qubits which are involved in the computation. For all relative-phase gates, a slightly different detuning energy is chosen for each qubit and all qubits are off-resonance from each other. Entangling between the qubits is done when all qubits that are to be entangled are detuned to the same energy A.S2. The error in energy-transfer gates

[0173] The energy -transfer gates between the cavity and the qubits are here analysed when taking Aj = 0 for all qubits so that H= g 1X1(81—+ s / n~ For N = 1, a perfect X gate (fidelity of 1) between the cavity and the qubit are supported in this architecture given a single value of n in the cavity quantum battery is required) when applying the interaction for Notably, a shorter gate may cause a Xagate so that the final state is the superposition state (1 — sfa 1 O n) + Va| lp n — 1), which should enforce entanglement with the next operations (since the gate is dependent in n). This connection prevents us from proving rigorously that the shared-cavity structure can support a universal quantum gate-set, but open up unique gates such as controlled-unitary gates

[0174] When the number of photons in the cavity has a single value, a perfect multi-qubit energy-transfer unitary (X®w) can be executed with N steps, where the duration of each step is determined by the number of quanta in the cavity before the step. Then, the total charging time should be TfuU charge=. When n » N, TfuU chargegrows linear with N in zeroorder. A faster gate can be reached when executing parallel energy -transfer gates but with compromised fidelity.

[0175] To reach the errors in these superextensive energy-transfer gates, a simplified case where n » N may be analysed. Then, the following approximation is made ^Jn — nq= / —A AA1 / 2~ / — i p-.= fn — leading toN NHint= d^n Gt ~ J^q + Jfiq°d,i) ■ (S8i=l i=l

[0176] The first term of Eq. (S8) corresponds to the local, yet superextensive, parallel X gates. The superextensive power exchange arises since the gate duration is t = = 7^= whichreduces as N even though the ratio r = n / N remains constant. The second term in Eq. (S8) corresponds to the first-order gate error.

[0177] The average gate fidelity is derived asFr= — 22N | \TlrrfV[U7i+deal Uuacrtua ^l Jl \2= — 22Nl rleIwhere the unitary gates are substituted, Uideai= XNand Uactuai=hr^Si=1l+y]nqad,iX®Ne \ / when applying the Hamiltonian in Eq. (S8) for t = — =. By 2g n continuing with the approximations,where the first-order term of the exponent nullifies since its diagonal terms are 0. Next, the trace term is examined asN= Tr ((iVe- l)iVe+ (N - Ne)Ne) ( / V - l)Tr = (Ne) = (IV - 1) k Q) fc=0= (IV — 1)7V2JV— 1where the following is used < Jt Ne= NgO^a-l = ( / Ve— l)o'+. Therefore, F = 1 —1)A 1 — 2, and since N « n, r «11 « - the average gate fidelity becomes(S9)ideal U actual) | — 1 2

[0178] Therefore, for the first order, the reduction in collective gate fidelity is determined by rand goes like 3lrz. However, the error per single gate goes like reaching the improvementwith N when keeping the ratio n / N fixed. Notably, the approximation for the fidelity is correct only for cases where n » N and numerical analysis show that the error actually grows with N, even if r remains unchanged. Still, it may be inferred that reaching above 0.99 fidelity for charging all qubits with a single gate requires r > 6. This requirement is relieved if several steps of charging are performed where less qubits are involved each step. An optimized circuit can reach a good trade-off between the fidelity, the overall gate time, the number of operations, or the heat which is created to support the computation.S3. Towards entangling gates, converting to the angular momentum basis

[0179] To reach an analytical solution to the entangling collective gates, it is convenient to transform Eq. (S6) to a Hamiltonian with collective operators where all involved qubits are detuned to a similar A. The collective operators are Jx,y,z= which satisfy [J1, / 7] =where Stjkis the Levi-Civita tensor, and J- = Yi=i <?a,i=iXi=2( / x+ i]y). The flip between + and — in the last equality arises from the notation that |0)®w=| / = rrtj = H so that 0+ adds an excitation to the system but reduces m.j. Additionally,and Eq. (S6) becomes(S10)

[0180] Now it may be seen that |J, m J) is the natural basis of the Hamiltonian in Eq. (S10), where - J < mj < J, Jz\j>mj) = mj\j, and ]±\], + m)(j + m + T) \j, mj ± 1). Notably, J2remains constant under the Hamiltonian in Eq. (10), though it can get any value of 0 < J < according to the state of the qubits prior to the similar detuning. This mixture of J values may be the source of the entangling gates.To further simplify the following derivations, the operator may be defined asso that A\j,mj) = H n — I + Jz\j,mj) = H n — + mj\j,mj) and [A,fz] = 0. Whensubstituting these operators to Eq. (S10), and removing a global phase of — A-, the Hamiltonian becomesH = ]z+ g(]~A + AJ+). (Sil)

[0181] With this Hamiltonian, the entangling gates which arise in the dispersive regime when A » g may be analysed.S4. The Schrieffer- -Wolff Transformation

[0182] Entangling gates are implemented in the dispersive regime. To get a better understanding of what gates are native the dispersive regime and the Schrieffer-Wolff Transformation is performed. Under the Schrieffer-Wolff Transformation for a Hamiltonian in the form of H = Ho+ V where | | « |H0|, the Hamiltonian of the system is approximated to H'es= (Ho+ V)e~s= HQ+ V + [S, Ho+ F] + 1 [s, [S, / ?]] + ••• = Ho+ 1 [S, F], where Ssatisfies V + [S, Ho= 0. To find S, the ansatz Sa = (j~A — AJ+>) is used,[S, Ho] = a j-A - AJ+, JZ] = aA [j-, Jz]A - A[ / +, / z]) = a(j~A + AJ+)where [A, JZ= 0 and [ / ±, / z] = +J±are used. To reach [S, Ho= — V a = —is chosen so that S = (AJ±J~A. Now, the commutation relations are calculated as2 2 [S, 7] = ^AJ±J-A)[g, (j~A + AJ+)] = 2^ [AJ+tJ-A] = 2 ^ (A]+]~ A - J~ A2J+).

[0183] Since J+J~ commutes with Jz, it also commutes with A, leading toAJ+J~A = A2J+J~ = in - -) l + JZ} J+J~ = [ n — — J+J+JZJ+J~.

[0184] In addition, J~A2J+= (n — + J~JZJ+so thatA / + / -A -J~A2J+= [n - + JZJ+J~ - [n - ~J~JZJ+ / N\ / N\=\n~ 2 )+- 2 )2 / z + / N\ 2 =2(n - -J / z+2( / z) - J~J+

[0185] So that the overall dispersive Hamiltonian becomesHd= H0+ - [S, V] = AJz+ ^- [2 [n - -)Jz+ 2(jz)( q2 / 1V\\ ~ 2q2.2 q2~ ~ = ( A + 2 ^ (n - -) Vz+ -|- ( / z) - y / 7+(S12)which is similar to previous derivations. Analysing this approximated Hamiltonian From Eq. (S12), although not exact, can give a good understanding of the possible gates, and show how thecombination of n, the ratio —, and the gate duration, determine the executed gate.S5. The Controlled-unitary protocol

[0186] So far, the claim-to-fame of the shared-cavity architecture was its capability to prepare a n-qubit GHZ state with a single entangling gate. However, a notable property in being able to execute QEC protocols is that the number of photons in the cavity (the quantum battery) may determine the collective gate which is applied to the qubits.

[0187] By applying a X1 / 2gate to one of the qubits (an ancillary qubit) when the cavity has n photons, the ancilla-cavity state should be -= (|0a)® |n) + | la)® |n — 1)). Then, by detuning the ancillary qubit, and applying a gate to the rest of the qubits, the entangling unitary I7entbecomesU = ^ (|0a)(0a|®[7ent(n) + | la)(la|®[ / ent(n - 1))= f i / ent(n) (|0a)(0a|® / + |la)(la|®(t / ent(n))-1t / ent(n " 1)) ■ (S13)which is a controlled gate. Notably, if the computational qubits’ state is an eigenstate of U = (t / entG'i)) Uent(n — 1) than the ancillary qubit may get a phase according to the eigenvalue of that eigenstate. This effect is also called a phase-kickback which is used in many quantum algorithms. Thus, the system may probe the eigenstate of any unitaryUprobe ^local ent(^)) ^entOt ~ 1)^ ^local (S14)where t / localisaunitary gate which does not involve entanglement (but can be collective). To probe the unitary, another local X1^2gate is applied to the ancillary qubit and measure it. If the measurement outcome is 1 (or 0), it is known that the computational quantum state collapsed to a subspace defined by the eigenstates of Uprobewith eigenvalue 0 (or 1). Lastly, even if the computational quantum state was not originally an eigenstate of the probed unitary, a measurement-based gate may be applied to the computational qubits to reach a required state.

[0188] In QEC, probing a specific set of stabilizers (joint-Pauli operators) is a required building block. The architecture facilitates this requirement since any gate in the dispersive regime keeps the number of qubit excitations. This means that [Z®w, I7ent]=0 foranYn- Specifically, when substituting Eq. (S12):- 1) = e2it(S15)

[0189] To reach an exact equality (up to a global phase) of Uprobe= el*Z®w, the gate is applied for t = -^7, to reach Uvrobe== (— 1)^ = (— 1)(+We) = iNZNwhich is exactlythe gate to be probed. Therefore, the system can probe any Z®w(for any N) with a single collective gate, which does not depend on the cavity’s state. When including local operators, may be potentially probed, which completes the stabilizers for any QEC stabilizer code. Notably, applying the gate for a different time can enable a different controlled-U gate which might be beneficial for other quantum algorithms.S6. The contribution of Counter Rotating Terms

[0190] The derivation presented earlier in this disclosure was restricted to the rotating-wave approximation (RWA), where the system Hamiltonian is block-diagonal in the excitation number basis and all terms conserve the total number of excitations. Relaxing this approximation introduces in Eq. (1) additional counter-rotating terms of the form#CR = ) > (S 16)which couple states belonging to different excitation-number sectors. Physically, these terms enable processes that simultaneously create or annihilate excitations in both the qubit and the resonator, thereby breaking the conservation of the total excitation number. Such processes become relevant when the coupling strength g approaches the characteristic transition frequencies co of the system. In this regime, the rotating-wave approximation is no longer strictly valid, and the dynamics can deviate from those predicted by the block-diagonal Tavis-Cummings description.

[0191] To quantify the impact, simulations including the full Hamiltonian with counter-rotating terms were performed, as shown in Fig.18. When keeping the detuning parameters exactly as in Fig. 15e, the infidelity grows systematically with the ratio g / (), reaching the percent level already at g = 0.003® (corresponding to g = 15 MHz for a resonator frequency of 5 GHz). Nevertheless, a modest re-optimisation of the detuning times, while keeping the detuning amplitudes fixed, is sufficient to suppress the counter-rotating errors by nearly two orders ofmagnitude for specific working points of g I a>. This correction only requires small adjustments of the timestep to compensate for relative phase accumulation induced by the counter-rotating terms. The effectiveness of such a minimal correction indicates that the disclosed system is robust against moderate deviations from the RWA regime. However, when the coupling strength exceeds the percent level of the transition frequency, further parameter re-optimisation, which go beyond simple time adjustments, may be used to maintain high-fidelity operation.

[0192] Fig. 18 illustrates the effect of counter-rotating terms. Infidelity of the circuit from Fig.16 when counter-rotating terms are included. The blue curve corresponds to the original detuning parameters Fig. 16e, showing on a log-log scale how the infidelity increases with the coupling ratio g I c, reaching an: 1% error once the ratio reaches g = 0.003®. The other curves show how a re-optimisation step of only the original detuning times for the full non-RWA Hamiltonian can reduce infidelities by almost two orders of magnitude, when provided a target g! co value (green: g = 0.0008®; orange: g = 0.003®; red: g = 0.01® ). The re-optimisation step modifies the timestep by a small factor, only to overcome the relative phase accumulation due to the counter rotating terms. Further error-reduction when gl co > 1% will require a more significant parameter re-optimisation.

[0193] Overall, from these simulations it is clear that the counter-rotating terms do not prevent quantum computation with quantum batteries. When g / co is below 1%, which is expected in typical superconducting circuit hardware (e.g., g < 50 MHz), a small correction to parameters derived from the RWA Hamiltonian is sufficient to suppress the contribution of the counterrotating terms. For g / co values above 1%, a shared resonator can still provide energy to the computational qubits, although a more significant parameter optimisation is required, and the block-diagonal description (including the dressed-operator representation) can no longer be employed.S7. Heat dissipation analysis

[0194] The goal here to compare the conventional quantum computation to the shared-cavity computation in terms of the created heating power in the bottom two stages of the quantum computer. Then, how many qubits can a standard cryogenic fridge support may be derived, provided the shared-cavity computation. Here, the lowest two layers in the cryogenic fridge (cold plate, CP, and mixing chamber, MXC) are focussed on, as these are the notable ones to limit thenumber of qubits in the fridge. The percentage of cooling power may be analysed, which is needed to operate the quantum computation in a cryogenic fridge (e.g., Bluefors XLDlOOOsl).

[0195] Channel types for running a quantum computation

[0196] The channel configuration is oriented to flux -tuneable superconducting qubits, though similar analysis can be done for fixed-frequency qubits and semiconductor spin qubits. In this configuration there are five types of channels to operate a standard quantum computer:

[0197] 1. Drive channels - Controls the XY gates with analogue modulated pulses in the GHz range through a coaxial line which is capacitively coupled to the qubit. Each qubit is the system has a unique drive channel, and full control over XY gates require two drive channels at room temperature (and also a mixer). These channels are used for all local (single-qubit) unitary gates. In each one of these gates, energy is always pumped into the fridge. Notably, these channels include large attenuation (20 dB in both the MXC and CP) to minimize the thermal noise and thus cannot use superconducting wires. Therefore, these channels create a significant portion of the overall computation heat, without any significant prospects in reducing this heat (may remain with stainless-steel materials) in a standard qubit configuration. The shared-cavity suggestion removes the need for these channels and thus can reduce significantly the power in running quantum computation.

[0198] 2. Flux channels - In superconducting qubits, the flux line is a current line which determines the flux through the Josephson Junctions within the qubit. Each qubit in the system has a unique flux channel which controls the qubit’s resonant frequency using low-frequency analogue signals. These channels use a stable static DC current to determine the operating point of the qubit. Flux-tunable qubits use analogue signals of 1-2 GHz for entangling fast-flux gates. In terms of heat loads, these channels have a single attenuator in 4K, enabling the choice of superconducting wires for a full operation. In such a scenario, the heat created by the DC current and switching becomes negligible.

[0199] 3. Readout resonator drive channels - These channels are used to apply an analogue microwave signal to a resonator which is in dispersive coupling to a qubit. These pulses can read the state of several qubits in parallel via frequency multiplexing (assuming 1 channel per 8 qubits). Resonator drive lines include similar attenuation as the XY drive lines but has a signalamplitude around an order of magnitude lower than the XY signal. Thus, their passive heat is similar to a drive line but their active heat is negligible.

[0200] 4. Readout output channels - Output lines use superconducting wires to reduce the noise as much as possible, and do not include attenuators. There is one output line per resonator drive line.

[0201] 5. Readout Amplifier Pump channels - To amplify the readout signal, microwave pumping a cryogenic amplifier in the MXC (usually a traveling wave parametric amplified, TWPA) is used. These channels include 10 dB less attenuation in the CP, but may use significantly more power (-55 dBm in the MXC). Each readout line uses a unique amplifier and thus a pump channel. Overall, the active load of the pump channels was found to be comparable to the drive lines (larger amplitude but less attenuators).

[0202] Cable types and configuration

[0203] 1. Stainless steel coaxial cables (UT-085-SS-SS), chosen for all channels which include attenuators in the CP and MXC.

[0204] 2. Niobium-titanium cables (UT-085-NbTi), which can potentially be chosen for all channels without attenuators in the CP and MXC. The passive loads from the theoretical calculation are taken, assuming that the NbTi indeed reached the superconducting phase. In addition, it is assumed that the active loads in the superconducting phase can be taken to zero.

[0205] 3. Optimal superconducting cables which the twisted-pair wires are analysed.

[0206] In this derivation, the heat created during the quantum computation between the standard architecture, the shared-cavity architecture with standard cables, and shared-cavity architecture with superconducting cables are compared. Table SI summarizes the cable configuration for these three architectures.Drive Flux Resonator drive output Pump Standard SS SS SS NbTi SS Shared-cavity SS SS SS NbTi SSStandard w. SC SS opt SS NbTi SS cablesShared-cavity w. SS opt SS NbTi SS SC cablesTable SI: Cable con iguration for the three analysed architectures. The ca )le types relate to Stainless steel coaxial cables (SS), Niobium -titanium cables (NbTi), and optimal superconducting cables (opt).

[0207] Heat sources

[0208] There are two fundamentally different sources of heat, passive load and active load.

[0209] Passive load: Passive heat load corresponds to heat which is created from thermal fluctuations. Specifically, the thermal fluctuations in room temperature which are conducted to all stages in the fridge (via to the cables that have a non-zero thermal conductivity). The passive heat is created along the cables and within the attenuators due to the thermal resistivity, calculated from the cable materials and dimensions (cross sections and length). In this analysis the calculated passive heat load per cable is used, and shown in Table S2. It is noted that the passive heat from drive and flux lines are different due to the different attenuator configurations. The optimal SC wires passive heat is taken from the twisted-pair dataCP MXC UT-085-SS-SS (SS); Drive 365 nW 8.5 nWUT-085-SS-SS (SS); Flux 270 nW 17 nWUT-085-NbTi (NbTi) 240 nW 11 nW Optimal SC wire (opt) I nW 0.01 nWTable S2: Calculated passive heat of each cable type for the two lowest stages.

[0210] The total expected passive heat per qubit in each configuration is shown in Table S3. The cable configuration from Table SI is used in addition to the expected multiplexing in readout line and the passive heat from the attenuators. For the shared cavity configurations, one drive line per 10 qubits is kept to account for the lines used for charging the cavity. This additional line adds passive heat and not active heat since the battery charging is done prior to the computation.CP MXC Standard PP / CP= 756 nW Pp. Mxc=29 nW Shared-cavity PP, CP = 427 nW Pp. Mxc=21.35 nW Standard w. SC cables Pp Cp = 487 nW Pp. Mxc=12 nW Shared-cavity w. SC cables Pp CP= 158 nW Pp. Mxc=4.36 nWTable S3: total passive heat per qubit for each option, given the cable configuration above. The standard case is calculated through 1.25 SS drive cables, 1 SS flux, and 0.125 NbTi cables per qubit. The shared cavity case is calculated through 0.35 SS drive cables, 1 SS flux, and 0.125 NbTi. The standard with SC cables case is calculated through 1.25 SS drive cables, 1 opt cable and 0.125 NbTi cables per qubit. The shared-cavity with SC cables case is calculated through 0.35 SS drive cables, 1 opt cable and 0.125 NbTi cables per qubit.

[0211] Active heat: Active loads arise from the external power which actively enters the system and is dissipated along its way inside the cryogenic fridge. The heat dissipation occurs (i) along the cables due to cable resistivity and (ii) within the attenuators. Attenuation is mandatory to reduce the incoming noise radiation field. That is, attenuators are needed to block the high occupation of room-temperature photons to reach the quantum computer and modify the quantum state. As a result, the input powers of the drive signals in room temperature are significantly larger than what are used to interact with the qubits. The active heat for each one of the channels is broken down in the following:

[0212] 1. Drive channels - The drive lines include 60 dB attenuation along the cable (photon attenuation of 106), with 20 dB are in the MXC and 20 dB in the CP. The average power per 20 ns n - pulse as Pavg(Ti) = —71 dBm (= 8 • 10-11W) and a TT / 2 pulse is Pav5(ii / 2) =—77 dBm may be calculated. The average powers over time may be further multiplied by the finite duty cycle of these pulses, 0 < D < 1, during the execution of a quantum algorithm. When taking the surface code QEC cycle (and include dynamical de-coupling) as an example, using D = 0.2 per qubit on average, when including the cycle gates and dynamical decoupling.Overall, the average power needed to interact with the qubit is Pavg= D ^Pavg( + Pavg Q))so that drive channel active load per qubit that were used are Pa, Mxc= 2Pavg=2.5 nW; P^cp= 104Pavfl, = 250 nW.

[0213] The shared-cavity cases do not create any active heat in the drive lines which are used for charging the battery prior to the computation.

[0214] 2. Flux channels - Dissipation in flux lines is mainly sourced from the DC biasing currents, which are constantly applied to set the qubit frequency. The worst case-scenario may be0.050 pW and 0.140 pW on the MXC and CP, respectively, per channel. However, when adding another ~ - overheat for entangling gates, but reducing a factor of 3 when assuming better than worst case scenario (careful magnetic shielding), the overall active flux power per qubit may be P£MXC=20 nW;CP= 54 nW.

[0215] These numbers correspond to the DC resistances of the stainless-steel cable (noticing that the cable ends with 50 Ohm to calculate the RT power). Using superconducting cables, the flux lines active load could be brought to close to zero. Therefore, for the optimal scenario of superconducting cables the active heat due to the flux channels is nullified.

[0216] 3. Readout resonator drive channels - These signals are typically an order of magnitude smaller than the qubit drive channels. When assuming a similar duty cycle as the qubit drive pulses, but one pulse per 8 qubits, it is found that the power per qubit due to this source is PaDMxc = 30 pW and P*DCP= 3 nW

[0217] 4. Readout output channels - do not create active heat.

[0218] 5. Readout Amplifier Pump channels -The power level at the input of the TWPA is used to be about -60 dBm, and a duty cycle of 10% is used. The channel includes 50 dB attenuation along the way, with 10 dB attenuation in the CP and 20 dB in the MXC. Thus, taking one pump line per 8 qubits, the heat per qubit used is PMXC= 2.5 nW and PCP= 25 nW.

[0219] Total heat power per computation

[0220] Table S4 summarizes the sum of active and passive heat per qubit per configuration for each cryogenic stage. The total number of qubits per fridge is derived by taking these heating powers and compare them to a cryogenic fridge (e.g., Bluefors XLDlOOOsl) colling powers of PCOOI, MXC = 34 iW and PCOoi,cp = 1000 1W in the MXC and CP stages, respectively. When taking the minimum in the number of potential qubits in the CP and the MXC it is found the overall qubit limit of the cryogenic fridge per configuration, shown in Table S5. In Fig. 19, it isshown the part of the different heat sources in the overall cooling budget. The limit comes from the MXC and not the CP stage, since cryogenic fridges have improved the CP cooling power by a factor of 5 while the MXC cooling power was improved by a factor of 1.7.CP MXCStandard Pex. cP = 1024 nW Pex. MXC=52 nW Shared-cavity Pex, cp = 497 nW Pex. MXC=42 nW Standard w. SC cables Pex. cp = 701 nW Pex. MXC=15.2 nW Shared-cavity w. SC cables Pex,CP= 173 nW Pex. MXC=5.6 nWTable S4: Total heat per qubit for eac i option, given the cable configuration above. The total power includes the sum of Table S3 with the calculated active heat.CP MXC limit Standard 976 657 657 Shared-cavity 2011 808 808 Standard w. SC cables 1426 2226 1426 Shared-cavity w. SC cables 5755 6033 5755Table S5: Qubit limit per configuration.

[0221] Computation energy analysis

[0222] The power that is calculated here is based on the active heat in the CP, multiplied by all the attenuation which connects the CP to room temperature (RT). A cycle of 1 ps is considered, where each gate is performed according to the mentioned duty cycles above. The power per cycle is calculated and multiplied by the number of cycles to get the total energy. The flux has additional 10 dB, the pump 20 dB, and drive (and readout drive) 20 dB compared to the CP level. For the shared cavity, there is additional energy overhead, which was determined as the power of 100 rounds of average drive power (equivalent to 10 drive rounds per qubit due to multiplexing).S8. Universal gate-set in the quantum battery system

[0223] In a quantum battery framework, the system naturally implements entangling gates and can perform Z rotations of arbitrary angles by detuning a single qubit relative to the others. Owing to the structure of the Lie algebra of SU(2), realizing a universal gate set relies on theability to implement a local non-trivial energy-changing gate — an operation that modifies the population of one qubit without influencing the states of the remaining qubits, regardless of the system’s state immediately prior to the operation, such a gate alters the occupation probabilities of the |0) and |1) states without inducing a full population inversion (applying the gate on |0) may not reach 11) and vice versa). Examples includde a Hadamard gate, [X gate, Ka(with a n, 2n), or a IT rotation around any axis outside the XY plane.

[0224] Without loss of generality, assume that the system can implement an Xa= e-lXaiTgate. If a is irrational, repeated applications of Xacan generate all rotations around the X axis when applied for n time (using the appropriate n values for the target angle). Combining this gate with an arbitrary rotation (P) about the Z axis, enables one to realize a Yarotation or more generally any rotation of angle a around an axes in the XY plane since Z^XaZ~^ = cos(2p) Xa+ sin(2p) Ya. If a is rational, the flexibility of arbitrary Z rotations enables the synthesis of an effective irrational rotation by appropriately interleaving the rational and irrational gate angles.

[0225] To illustrate the implementation of such a gate, a system composed of two qubits and a quantum battery initialized with n photons (with all qubits initially in the ground state) is initially used. One qubit is set to zero detuning (Ao= 0), while the other is highly detuned from the cavity (Ax» g) this ensures that during the gate operation, the highly detuned qubit undergoes only a phase change, leaving its population unchanged. In this configuration, the subsystem comprising qubit 0 and the battery shares n quanta when the state of qubit 1 is |0) or n — 1 quanta when the state of qubit 1 is 11), resulting in a Rabi frequency of Q.n= 2g\ / n or = 2g^ / (n — 1), respectively. To implement an identical gate operation in both cases, the following condition may be satisfied:(lnt = + 2TTJ1 71where j is a non-zero integer. Choosing a gate duration of t = - (j = 1) may execute onqubit 0 the Xagate with a = > regardless on the state of qubit 1. It is noted that for thistwo-qubit system n > 2 is used to span all qubit states. The non-trivial angle a thereby ensures that the implemented gate is sufficient to complete the universal gate set for the two-qubit system.

[0226] This logic can be extended to systems with more qubits. In such cases, besides selecting the appropriate gate duration, one can also exploit additional degrees of freedom — specifically, the detuning value Aoand the number of discrete steps (i.e., different detuning values applied over different time intervals). For instance, consider a three-qubit system with a quantum battery initialized to 5 photons. By applying detuning values of 6.5g and —6.76g to one qubit for durations of 24.13 and 24.54 respectively, the system implements a 0.96n rotation about theaxis (0.07,0.811,1) when the other two qubits occupy the { 101), 110)} subspace. If those qubits are instead in the 100) or 111) states, the process fidelities (evaluated relative to the {| 01), 110)}1 + 2subspace using F([ / 1, 1 / 2) = j |Tr(V1Tl / 2)| ) are 99.2% and 98.9% respectively, resulting an average gate fidelity of 99.5%.

[0227] A numerical optimisation is used to determine the optimal detuning values and durations for executing a local unitary operation in systems comprising 3, 4, and 5 qubits, with the battery photon number n reaching up to 7. As shown in Fig. 20, the worst-case fidelities exceed 99.5%, and the average fidelities surpass 99.8%, when using only two detuning steps. This procedure can be readily generalized to systems with an arbitrary number of qubits, thereby enabling the implementation of a local, non-trivial energy-changing gate and, ultimately, a universal gate set.S9. Simulation details

[0228] All quantum simulations were performed using Python software with the Qutip package. For the superextensive computation in Figs. 12a and Fig. 12b, an extended Hilbert space is used which includes both the light and the qubit states while all other quantum simulations include a reduced Hilbert space of dimension 2Waccording to the chosen computational block. In Figs. 12a and 12b, the detuning of all qubits was 0, and the gate duration was determined as the time which reaches the optimal final state fidelity compared to |1)®W. In the QEC simulations of Fig. 16, the detuning values and the gate times were optimized for each gate throughout the circuit to reach the highest fidelity to the quantum state after each gate in Fig. 16. In cases of local gates, optimisation of detuning values and gate time of more than a single step was enabled, until reaching the maximal fidelity. It was found that knowing in advance the expected optimized values, according to the analytical derivation, enabled a significantly faster convergence of the optimisation procedure of which the Scipy package was used.

[0229] Fig. 19 shows heat power source distribution for the examined computation architecture. Fig. 20 shows the fidelity of the non-Pauli energy-exchange gate as a function of the total number of photons in the battery for different multi -qubit systems. The shaded circles represent the fidelities given different initial multi -qubit states. For systems with three or more qubits, the average gate fidelity remains above 99.8%, while the worst-case fidelity exceeds 99.5%, when using a two-step detuning process. In the two-qubit case, a single detuning step achieves 100% fidelity with a single detuning step (shown analytically in Appendix section S8).

[0230] It will be appreciated by persons skilled in the art that numerous variations and / or modifications may be made to the above-described embodiments, without departing from the broad general scope of the present disclosure. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive.

Claims

CLAIMS:

1. A quantum computing system comprising:a quantum processor comprising multiple qubits, the quantum processor being configured to implement one or more operations on the multiple qubits; andan energy storage system configured to store energy in a quantum state,wherein the energy storage system is coupled to the multiple qubits and configured to supply the multiple qubits with energy from the quantum state to perform the one or more operations.

2. The quantum computing system of claim 1, whereinthe quantum computing system further comprises a control element configured to create entanglement between a quantum state of one of the multiple qubits and the quantum state of the energy storage system; andthe quantum state of the one of the multiple qubits remains entangled with the quantum state of the energy storage system after performing one of the one or more operations on the multiple qubits.

3. The quantum computing system of any one of the preceding claims, wherein the quantum state of the energy storage system is a bosonic mode.

4. The quantum computing system of any one of the preceding claims, wherein the energy storage system is configured tocreate a qubit excitation by annihilating of quanta in the energy storage system; and annihilate the qubit excitation by creating of quanta in the energy storage system.

5. The quantum computing system of any one of the preceding claims, wherein the quantum state of the energy storage system is a Fock state.

6. The quantum computing system of any one of the preceding claims, wherein the energy storage system comprises a resonance cavity for electromagnetic waves and the multiple qubits are coupled electromagnetically to the resonance cavity.

7. The quantum computing system of any one of the preceding claims, wherein the quantum computing system further comprises control channels to tune a resonant frequency of one or more of the multiple qubits.

8. The quantum computing system of any one of the preceding claims, wherein the multiple qubits are superconducting qubits and the quantum computing system comprises flux lines configured to provide flux control of the multiple qubits.

9. The quantum computing system of any one of the preceding claims, wherein the energy storage system comprises multiple energy storage systems each associated with a subset of the multiple qubits.

10. The quantum computing system of any one of the preceding claims, wherein the energy storage system comprises a Fock state larger than or equal to a number of the multiple qubits.

11. The quantum computing system of any one of the preceding claims, wherein a ratio of a size of the quantum state of the energy storage system and a number of the multiple qubits is between 1 and 2.

12. A method for operating a quantum processor, the method comprising:initialising an energy storage system into a quantum state; andperforming one or more operations on multiple qubits of the quantum processor by supplying the multiple qubits with energy from the quantum state.

13. The method of claim 12, wherein performing the one or more operations comprises setting a detuning frequency corresponding to a difference between a resonant frequency of one or more of the multiple qubits and a resonant frequency of the energy storage system.

14. The method of claim 13, wherein performing the one or more operations comprises performing a local gate on one of the multiple qubits by performing multiple tuning steps to at least one of the multiple qubits.

15. The method of claim 13 or 14, wherein performing the one or more operations comprises performing an energy transfer gate on one of the multiple qubits by setting the detuning frequency of the one of the multiple qubits to about zero.

16. The method of claim 15, wherein performing the energy transfer gate entangles a quantum state of the one of the multiple qubits with the quantum state of the energy storage system, wherein the quantum state of the one of the multiple qubits remains entangled with the quantum state of the energy storage system after performing one of the one or more operations on the multiple qubits.

17. The method of any one of claims 13, wherein performing the one or more operations comprises performing a relative phase gate by setting the detuning frequency to a different value for each of the multiple qubits.

18. The method of any one of claims 13 to 17, wherein each of the one or more operations is a unitary gate performed over a gate time, wherein the unitary gate is determined based on the gate time and the detuning frequency.

19. The method of any one of claims 13 to 18, wherein performing the one or more operations comprises performing an entangling gate on a subset of multiple qubits by setting the detuning frequency for each of the subset to a similar value.

20. The method of any one of claims 13 to 19, wherein performing the one or more operations comprises performing a collective gate on the multiple qubits by performing one or more detuning steps for each of the multiple qubits.

21. A method for determining an ancilla qubit state in relation to multiple data qubits, the method comprising:tuning an ancilla qubit to transfer energy from a quantum state of an energy storage system, to entangle the ancilla qubit with the quantum state of the energy storage system resulting in a changed quantum state of the energy storage system;tuning the multiple data qubits to transfer energy between the multiple data qubits, wherein the transfer energy between the multiple data qubits is based on the changed quantum state of the energy storage system to create a relationship between the ancilla qubit state and a parity of the multiple data qubits;tuning the ancilla qubit to transfer energy between the ancilla qubit and the energy storage system with a direction based on the parity of the multiple data qubits; andmeasuring the ancilla qubit to determine the ancilla qubit state, the ancilla qubit state reflecting the parity of the multiple data qubits.

22. The method of claim 21, wherein tuning the ancilla qubit comprises tuning the ancilla qubit to disentangle the ancilla qubit from the changed quantum state of the energy storage system resulting in a pure quantum state of the ancilla qubit that reflects the parity of the multiple data qubits.

23. The method of claim 22, wherein the method further comprises, after tuning the ancilla qubit to disentangle the ancilla qubit, further tuning the multiple data qubits to restore an original quantum state of the multiple data qubits.

24. A method for creating a logical qubit for quantum error correction on a quantum computing system comprising an energy storage system, the method comprising tuning a resonant frequency of multiple qubits of a quantum processor using the energy storage system to entangle the multiple qubits into the logical qubit.

25. The method of claim 24, wherein the method further comprises performing a probing operation, wherein the logical qubit is in a superposition state after the probing operation, and performing the probing operation comprises performing a controlled-parity operation of a subset of the multiple qubits using an ancilla qubit of the quantum processor.

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