Superfluid-based Quantum Circuit Components

Abstract

A quantum circuit component 100 comprises: a reservoir 106 containing a first sample of a fluid 104B (e.g. liquid helium), wherein the fluid is in a superfluid phase when cooled below a superfluid cri

Description

Field This specification relates to components for quantum circuits. In particular, this specification relates to components for quantum circuits that are based on superfluids. Background Many platforms are currently used to realise qubits, including quantum circuits using superconductors, nuclear spins, photons, trapped atoms or ions, quantum dots, and photons. The most widely used platform for qubits is a superconducting quantum circuit. This is a circuit in the conventional electrical sense, with specific circuit elements that have inherent quantum properties. These "quantum" circuit elements / components enable the macroscopic quantization of the degrees of freedom of the circuit as a whole. The circuit then behaves as a highly controllable artificial atom, with well-defined quantized energy levels, whose properties can be engineered to function as a qubit. A superconducting quantum circuit uses a Josephson junction as the "quantum" circuit element. Depending on the circuit geometry, various superconducting quantum circuits realise a range of qubits such as the transmon, the flux qubit, etc. Linking adjacent superconductors through an insulating (non-superconducting) layer forms superconducting Josephson junctions. This comprises the basic "quantum" element of a superconducting quantum circuit. Summary According to a first aspect of this specification, there is described a quantum circuit component comprising: a reservoir containing a first sample of a fluid, wherein the fluid is In a superfluid phase when cooled below a superfluid critical temperature; and a cell within the reservoir comprising a second sample of the fluid, wherein the second sample of fluid is in a superfluid phase when cooled below the superfluid critical temperature. The cell comprises: one or more weak links between the second sample of the fluid and the first sample of fluid; and one or more displaceable mechanical objects coupled to the one or more weak links via the second sample of fluid. The cell is associated with a characteristic frequency that depends on properties of the one or more displaceable mechanical objects, properties of the one or more weak links and properties of the first and second samples of fluid. The characteristic frequency is associated with quantised nonlinear oscillatory dynamics of the superfluid. The cell is configured such that the characteristic frequency associated with is less than two pi times a magnitude of a superfluid gap associated with the superfluid phase of the fluid. The second sample of fluid Is In the same superfluld phase as the first sample of fluid when cooled below the superfluid critical temperature. The second sample of fluid has the same superfluid texture as the first sample of fluid when in the superfluid phase. At an operating temperature below a temperature that corresponds to the characteristic frequency of the superfluid quantum circuit, the superfluld quantum circuit has quantised and resolvable energy levels, applicable as a platform to realise a qubit. The quantum circuit component may further comprise an electrode configured to measure one or more properties of the cell indicative of a quantum state of the cell and / or to manipulate one or more properties of the cell to control the quantum state of the cell. The electrode may comprise a superconducting quantum interference device, SQUID, a superconducting qubit, or a superconducting resonator. The cell may further comprise one or more fixed walls comprising the one or more weak links. The one or more weak links may comprise one or more apertures with dimensions comparable to a coherence length of the fluid when in the superfluid phase. The one or more weak links may comprise an array of N>1 weak links, wherein the weak links in the array are separated by a distance that is greater than the coherence length of the fluid when in the superfluid phase. The weak links in the array of weak links may be configured to function coherently as a single Josephson junction. The cell dimensions may be less than a superfluid texture healing length of the fluid when in the superfluid phase. In the superfluid phase, the first and second samples of fluid may have a homogenous uniform texture across the weak links that connect the first and second samples of fluid. The characteristic frequency may be a characteristic frequency associated with the small amplitude regime of the quantised nonlinear oscillatory dynamics of the superfluid. The characteristic frequency, <ap, may be given by: ., 2m^klr Mp=~piA^h where ms Is a mass of quasiparticles of the fluid, ps is a fluid density of the fluid, k is a spring constant associated with the one or more displaceable surfaces, A is an area of the one or more displaceable surfaces and Ic is a critical current for the Josephson effect between the first sample of fluid and the second sample of fluid when in the superfluid phase. The quantum circuit component may be described below the superfluid critical temperature by a Hamiltonian, of the form: W = Ec(n — ng^ — Ej cos 0 where Ec Is a charging energy associated with the cell, Ej is a Josephson energy, n is a number of Cooper pairs in the cell, ng is a number offset caused by a constant pressure difference between the first fluid sample and second fluid sample, and 0 is a superfluid phase difference between the superfluid inside the cell and the superfluid outside the cell. The fluid may be helium-3. The fluid may be in the B-phase of superfluid helium-3 when cooled below the critical temperature. According to a further aspect of this specification, there is described a method of operating the quantum circuit component of any preceding claim, the method comprising: cooling the quantum circuit component to an operating temperature that is below the superfluid critical temperature; encoding a qubit state in the quantum circuit component using two energy states of the quantum circuit component; performing a quantum computation, comprising applying one or more quantum gates to a set of qubits that includes the quantum circuit component; and measuring, subsequent to the quantum computation, the state of the quantum circuit component. The operating temperature may correspond to a frequency that is less than the plasma frequency divided by two pi. According to a further aspect of this specification, there Is described a quantum circuit comprising one or more quantum circuit components as described herein. According to a further aspect of this specification, there is described a quantum computer comprising a plurality of physical qubits, wherein one or more of the plurality of physical qubits comprises a quantum circuit component as described herein. The examples described herein may be implemented to realise one or more of the following advantages. Superfluid-based quantum circuit components can be utilised to implement quantum circuits that are analogous to superconducting quantum circuits, but that are agnostic to electromagnetic noise. The superfluid platform eliminates electrical and magnetic noise as a source of decoherence, as Its qubit states do not respond to either charge or magnetic fields. The device has no electromagnetic intrinsic elements, thus providing unprecedented benefits In qubit operation. The oscillations of the devices are quantised at a well-defined resonance frequency, resolvable at low temperatures essential to the superfluid state. The superfluid-based quantum circuit components are highly scalable without loss of coherence from an intrinsic design perspective - a feature unavailable in other qubit platforms known thus far. The superfluid-based quantum circuit components described herein can be used to build qubits and other components in a quantum processor and, subsequently, a quantum computer that can potentially offer highly improved coherence times. These are essential to the performance of quantum computers and a limitation in current qubit platforms. The designs described herein are also scalable providing a direct route to realising quantum processors with several superfluid-based quantum circuit components. From a fundamental point of view, the devices / component described herein offer a scheme to realise the quantum regime of a mechanical macroscopic device. It is the first such scheme that realises such a regime using superfluids. It could potentially also be the largest scale (biggest in size) such device than can be cooled to its quantum ground state. The devices / components described herein could provide a basic apparatus for studies of Josephson physics between different superfluid phases and / or same similar superfluid phase with dissimilar textures - this could be for both fundamental and applied benefits In the context of superfluid circuits designed using superfluid weak links. Brief Description of the Drawings Example Implementations will be described by way of reference to the accompanying drawings, in which: FIG. 1 shows a schematic overview example of a superfluid-based quantum circuit component; FIG. 2A shows an example of a first part of an oscillation cycle of a superfluid-based quantum circuit component in use; FIG. 2B shows an example of a second part of an oscillation cycle a superfluid-based quantum circuit component in use; FIG. 3 shows an example of a circuit diagram for a superfluid-based quantum circuit component; FIG. 4 shows a flow diagram of a method for operating a superfluid-based quantum circuit component; and FIG. 5 shows a schematic overview of a quantum computer. Detailed Description Some fluids, such as liquid helium, undergo a transition to a superfluid state at low temperatures when cooled below a critical temperature, Tc. The superfluid is characterised by dissipation-less flow, analogous to resistance-free flow of electrical current In superconductors. The superfluid transition opens up a gap in the spectrum of the fluid (quasi)particles similar to the superconducting gap for electrons in the analogous case. Linking adjacent reservoirs of superfluid helium can form a superfluid Josephson junction. The superconducting Josephson effect describes the tunnelling of Cooper pairs of electrons from one superconductor to the other through a non-superconducting thin layer. The thickness of this layer is set by the superconducting coherence length, i.e. is of the order of the superconducting coherence length. Massive helium quasiparticles (or Cooper pairs of 3He quasiparticles) cannot appreciably tunnel in an analogous sense. However, a small constriction connecting two reservoirs of helium can effect superfluid Josephson physics. The size of the constriction, often referred to as a Dayem bridge or "weak link", is set by the superfluid coherence length. Superfluid 3He flowing through nanosized apertures connecting two superfluid 3He reservoirs shows a periodic oscillating relation between mass current and the superfluid phase difference between the two sides, illustrating Josephson physics. The Josephson effect in superfluid 4He has been observed with nanoaperture weak links at higher temperatures close to the critical temperature of the superfluid. In this work, we present a quantum circuit composed of superfluid elements, e.g., superfluid helium-3 (3He) or superfluid helium-4 (4He). A priori, with the very small gap of 3He, this would seem impossible. However this specification describes the design and operation of a superfluid quantum circuit in the quantum regime. The circuit is designed to be operable in a regime where it maintains phase coherence for times long enough to function as a qubit. The device may be referred to as a Superfluid Helium Oscillator Quantum Device, the SHOQDevice. This specification further describes an example of using this device as a superfluid Cooper Pair Box (CPB) and the use of the device as a qubit. Such a device realises a quantum circuit in a charge neutral environment viz., a quantum circuit without electrons, charges or electrical currents; in other words, a quantum circuit that is not an "electrical" circuit. Consequently, the device is not susceptible to fluctuations in electromagnetic fields that can cause noise in superconducting devices. This can make the devices described herein good candidates for realising qubits, e.g., in a quantum computing system. Furthermore, the exotic properties of the superfluid phases (e.g., of 3He) can influence and potentially enhance the physics that underpins the devices described herein. The superfluid phases of 3He are a paradigm for spontaneous symmetry breaking and a model for unconventional pairing with p-wave spin triplet symmetry. They include topologically nontrivial phases that host exotic physics such as Majorana states, halfquantum vortices, anomalous quantum Hall effects and offer a testbed for cosmology in the lab due their rich order parameter structure. The conventional Josephson effect can be modified significantly in superfluid 3He. Additionally, the devices described herein could potentially be explored as a probe of this novel superfluid 3He physics. FIG. 1 shows an example of a superfluid-based quantum circuit component / devlce 100. In the following, the device 100 will mainly be described in the context of superfluid 3He-B, though other superfluids, such as superfluid 4He, may alternatively be used. The device 100 comprises a cell 102 containing a fluid 104A that becomes a superfluid when cooled below a critical temperature, Tc. The fluid 104A Is coupled to a reservoir 106 of the same fluid 104B via one or more apertures 108 that form weak links between the fluid inside the cell 104A and the fluid in the reservoir 104B. The apertures 108 have a size that Is of the order of the superconducting coherence length In some examples, a single weak link aperture 108 is used. In other examples, an array of N weak link apertures 108 is used. The cell 102 may be submersed in the fluid 104Bofthe reservoir. The cell 102 has dimensions that are larger than the superfluid coherence length, ^(T) is a function of temperature, T, approaching its maximum value = ^(T = 0). The dimensions of the cell 102 are large compared to ^o. In some examples, the cell dimensions of the cell 102 are also large compared to a characteristic size associated with confinement effects, e.g., larger than 10pm. In examples where an array of weak link apertures 108 is used, the apertures 108 are separated by a distance, d, that is larger than the superfluid coherence length, e.g., d >>^. The separation distance between apertures in the array is such that the apertures function coherently as a single Josephson junction, for example d~3pm. Such spacings can also act to minimise decoherence and dissipative effects. In some examples, tubes with all dimensions comparable to the superfluid coherence length may be used instead of weak link apertures. The tubes may have varied aspect ratios as long as each of the spatial dimensions of the tube is comparable to the superfluid coherence length. The cell 102 comprises one or more elastic and / or displaceable surfaces 110. For example, one or more of the surfaces / walls of the cell 102 may be a displaceable surface 110, such as a membrane or an elastic plate, e.g., a deformable elastic plate. The elastic and / or displaceable surface 110 displaces as the pressure inside the cell changes relative to the pressure outside the cell (i.e., the pressure in the reservoir). In some examples the one or more elastic and / or displaceable surfaces 110 may be a part or all of the lid and / or base of the cell 102. Alternatively, the one or more displaceable surfaces 110 may be a part or all of one or more of the side walls of the cell 102. In general, the cell 102 may comprise one or more elastic and / or displaceable mechanical components / mechanlcal oscillators, such as the one or more displaceable surfaces 110 shown in FIG. 1, one or more elastic beams, and / or one or more cantilevers. Many other examples are possible. For convenience, the quantum circuit elements described herein are described with respect to one or more displaceable surfaces 110, but may alternatively use other elastic and / or displaceable mechanical components. One or more of the other surfaces of the cell 102 may be rigid walls 112 that are fixed in position. The one or more rigid walls 112 are impermeable to the fluid 104A, 104B. The one or more apertures 108 are formed on one or more of the rigid walls 112. The rigid walls 112 may be curved such that the shape of the cell is a cylinder; this can allow the weak link arrays to be separated further while avoiding confinement effects. The one or more rigid walls 112 may form one or more side walls of the cell 102. Alternatively or additionally, the one or more rigid walls 112 may form a base and / or lid of the cell 102. The largest dimension of the cell 102 is less than a textural healing length, ^s, of the superfluid used in the device 100. The textural healing length is a length scale over which textures (i.e., smooth variations in the superfluid order parameter caused by, e.g., orientational forces such as the proximity of Cooper pairs to boundaries or surfaces) return to their undisturbed state. For example, in the presence of surfaces in 3He-B, the surface healing length fs(T)~0.271 - T / Tc cms. In containers much smaller than ^s, it is energetically more favourable for a texture to ignore the surface than to adjust its configuration close to the surface. Therefore, in cells 102 with dimensions much smaller than the textural healing length, the textural bending of the superfluid order parameter may be ignored, and a homogenous texture can be assumed. This is consistent with the assumption that the Josephson current is sinusoidal in the superfluid phase difference between the two sides of the weak link aperture(s) 108. Such a cell 102 can be achieved by carving out a cell-shaped cavity in a bulk stiff material, such as quartz or the like, which provides a rigid frame to which is attached a flexible plate / membrane. The flexible plate / membrane may be a thin wafer of a chosen material. Small displacements of the plate from equilibrium are linear in the pressure difference of the fluid between the two sides of the plate, i.e., the plate responds as a simple harmonic oscillator. In some Implementations, the device 100 further comprises a probe 114 for measuring and / or manipulating properties of the device 100. The probe 114 may be in the form of an electrode, such as a superconducting quantum interference device (SQUID) electrode, a superconducting qubit, or a superconducting resonator. The probe 114 may be configured to measure a quantum state of the cell 102. For example, the probe 114 may be configured to measure an energy of the cell 102. The probe 114 may be connected to further circuit elements / subsystems (not shown). In the example shown, the probe 114 is shown to be external to the reservoir 104. However, In other examples, the probe 114 may be directly coupled to the cell 102, e.g., to the mechanical oscillator and / or the rigid cell walls 112. The state of the cell could be manipulated / measured and possibly also prepared by coupling to a superconducting circuit, e.g., via the probe 114. Various schemes can be devised to couple quantised degrees of freedom of the superconducting circuit to quantised degrees of freedom of the superfluid quantum device component 100 / cell 102. For example, the elastic plate (or its equivalent) can couple capacitively to an electrical circuit giving a nonlinear interaction between the n-operator in the electrical circuit with that in the superfluid circuit. In another example, the quantum device component 100 / cell 102 can play the role of a capacitor in a superconducting circuit containing a Josephson junction. Here, the number of magnetic flux quanta in the superconducting circuit can couple to the number of circulation quanta in the superfluld circuit. Such coupling may be used to control, manipulate and / or prepare the state of the superfluid quantum circuit component 100 / cell 102. Such coupling can also be used to drive the superfluid quantum circuit component 100 / cell 102 to implement single qubit gates. In some examples, multiple superfluid quantum circuit components 100 / cells 102 share the reservoir 104 of superfluid. Such cells can remain entangled with possibilities to be applied along with single qubit gate operations to achieve multiple qubit gates. The cell 102 is associated with a characteristic frequency (referred to herein as a "plasma frequency"), cop, that depends on the properties of the fluid 104 (e.g., the density and / or effective mass of the fluid when in the superfluid phase), properties of the displaceable surface(s) 110 (e.g., the area of the surface(s) and / or a spring constant associated with the surfaces 110), and properties the weak link aperture(s) 108 (e.g., the size / area of the weak link aperture(s) 108 and / or the critical Josephson current associated with the weak link aperture(s) 108). In some examples, such as the example derived in relation to FIGs. 2A and 2B, the characteristic frequency of the cell is given by (e.g., approximated by): = a) where ms is the mass of the fluid quasiparticles (i.e., the mass of a Cooper pair In a fermionic superfluid is 2ms), ps Is the fluid density, k is the spring constant associated with the displaceable surface(s), A is the area of the displaceable surface(s) and lc is the critical current for the Josephson effect. However, it will be appreciated that other configurations may result in different forms of the characteristic frequency. The dimensions of the cell are configured such that wp / 27r is less than the magnitude of the order parameter / superfluid gap, A, of the superfluid in the cell and reservoir, i.e., When the temperature of the system, T, is less than —, this results in the cell behaving as a single coherent quantum object, i.e., the overall state of the cell can be in a superposition of quantum states, making the cell useful as a quantum circuit element. For example, for 3He-B, the superfluid gap can be estimated from the critical temperature of the superfluid (2.6mK at melting pressure) using BCS theory as: A(T = 0)~1.76Tc = 4.58mH 95.10MHz (3) The mass of the superfluid quasiparticles in 3He at the melting pressure is approximately ms = 3.12 x 10 26 kg, and the density of the 3He at the melting pressure is approximately ps = 118.09 kg / m3. These quantities typically vary little over the entire range of pressures. The critical current density is estimated to be ~1 kg m2s~l, giving a critical current of lc = aWLkg s-1, where awL is the weak link aperture area in m2. Under such conditions, the characteristic frequency becomes: „ 2m,klr , kaw, = —_—_— = 4.24 x 104—-r— (4) p psA2h A2 The main parameters available for engineering a characteristic frequency to operate the device in the quantum regime are thus the spring constant, k, set by the material used for the plate element (i.e. the displaceable surface), the size of the weak link apertures, which sets Ic, and the area of the plate, A. Values of the spring constant, k, displaceable surface area, A, and aperture size, awL, can be chosen to ensure that equation (2) is satisfied. As an example, typical weak link aperture sizes in superfluid Josephson experiments have an area of approximately aWL~100nm x lOOnm = 10-14m2 and a good mediumrange estimate for the value of k in materials used in superfluid weak link experiments is k-M^NnT1. Taking a circular disk of radius 8pm as the displaceable surface, this results in a value for the characteristic frequency of ~ 102.47MHz, and thus = 16.31MHz. Under such conditions, the eigenstates of the device Hamiltonian (derived below) are quantized with the ground and excited states separated by a frequency of cop / 2n ~ 16.31MHz. These levels are resolvable at a temperature of T = 0.4mK. The superfluid state is robust to excitations of energy <A and thus the quantum regime is attainable with such a device. The device 100 may be used as qubit, using the two lowest energy states as the computational states |0> and |1>. These states may correspond to the states with zero and one excess Cooper pairs in the cell 102. It will be appreciated that these values are provided by way of example only, and that other areas, spring constants and aperture sizes can also be used to satisfy equation (2). For example, the values used in Table I may be used. .V U AiOO ........ 4........... +..............— (if - ; 64 I 3 i : >b 95.10 102.47 1.30 il.97xWT 3.15 io" 10^ 25 j 34.36 95.10 51,8.46 33.28 7.7x10^ 9.10 10i$ w’ loOl 64 | 34.36 95.16 1924:.73 1.36 h.y~xiO!H 9.05 01" J.}2 33 ! 34.X 95.10 Hi 3.95 .1.3(} | 7.7 x 19s 0.32 10r 1F 1 i 25 j 21.0 726.21 9.11 i 1.49x10^ 0.07 Table I: Example parameter values for operating a superfluid helium-3 quantum circuit component in the quantum regime. FIGs. 2A and 2B show examples of the quantum circuit device 200 in operation. FIG. 2A shows an example of a first part of an oscillation cycle of a superfluid-based quantum circuit component in use, while FIG. 2B shows an example of a second part of an oscillation cycle a superfluid-based quantum circuit component in use. Complete oscillations illustrated cycle at the plasma frequency. FIG. 2A shows displacement 216A, hx, of the displaceable surface 210 (e.g., an elastic plate or membrane) from an equilibrium position 218 due to an inflow of Cooper pairs 220 into the cell via the weak link 208. FIG. 2B shows displacement 216B, Ax, of the displaceable surface 210 from an equilibrium position 218 due to an outflow of Cooper pairs 220 from the cell via the weak link 208. In use, the device 200 is cooled to an operating temperature, T, that is below the superfluid critical temperature of the fluid 204A, 204B. The fluid in the cell 204A and the fluid in the reservoir 204B are both in the superfluid phase at this temperature. In general the phase values of the superfluid order parameter in the two fluids 204A, 204B will be different, resulting in a superfluid phase difference, ¢, between the superfluids across the weak link 208. This phase difference causes a Josephson effect between the two superfluid samples 204A, 204B. Superfluid 3He flowing through the aperture(s) 208 connecting the two superfluid samples 204A, 204B shows a periodic oscillating relation between mass current and the superfluid phase difference between the two sides, illustrating Josephson physics. In general, the Josephson-like coupling between two samples of superfluid 204A, 204B depends on the nature of the order parameter in the samples on two sides of the weak link 208. For example, it would differ for linked samples that are in the same or different superfluid phases; or for samples in the same superfluid phase but with different textures on two sides of the link. However in all cases, this current-phase relation, including cases with textural dissipative effects, is nonlinear. In the following analysis it is assumed that the Josephson current is sinusoidal (i.e., I = lc sin 0, where I is the mass current and Ic is a critical current) and that the superfluid gap is of the BCS type; however, a similar analysis can be performed for other Josephson current relations and superfluid gaps. The role of surface roughness, disorder and other factors that affect the order parameter internal structure / orientation are disregarded here. The internal structure of the order parameters In samples 204A, 204B on both sides of the weak link 208 are identical and we explore the superfluid as a charge-neutral condensate of Cooper pairs. The Josephson-Anderson phase-evolution equation for the device 200 is given by = (S) dt h ' ' Where A / z Is the chemical potential difference and ¢ is the superfluid phase difference between the two sides of the weak link, t denotes the time variable. For a fluid, chemical potential variations are given by: m dP du =--S dT (6) P where m is the mass of the fluid particles with fluid density being p, dP is the pressure variation (i.e., the pressure variation between the inside and outside of the cell), S is the entropy and T is the temperature. Since the temperature variations in weak link experiments are known to be negligible and the mass of a Cooper pair is twice the mass, ms, of its constituent quasiparticles, equation (6) becomes 2mK All = ----AP. (7) Ps The pressure difference across the displaceable surface 210 will cause a displacement 216, x^t), of the surface 210 from its equilibrium position 218. The pressure difference is related to the displacement 216 via: k AP=-x(t) (8) / 1 where k is a spring constant associated with the displaceable surface 210 and A is the area of the displaceable surface 210. Using equations (5)-(8) the equation of motion for the phase difference can be rewritten as: 2m,AP 2m.k ¢ =--s—=-- Ps ft Ps ft A Any fluid entering the cell volume will displace the plate such that the mass current, I, is given by: I = ps A x, (10) which can be used rewrite equation (9) as: 2msk PsA2h Ic sin< / > = —Mp sin0 (11) where a sinusoidal Josephson relationship has been assumed. The motion of the displaceable surface 210 coupled to the weak link 208 is analogous to that of a rigid pendulum. If 0 represents a small angular displacement of a pendulum from the vertical, is the small angular frequency of oscillation. On timescales longer than the microscopic timescales of the superfluid, (e.g., for a fermionic superfluid h / eF, eF being the Fermi energy; this is ~ 10 ::sec in superfluid 3He), the change in energy of the superfluid in the cell is given by changes in the heat content, the mechanical volume energy and the mass, for a fluid that is not globally moving or rotating. The total energy, Etot, of this system / device is given by the energies stored in the plate Ep, that in the fluid in the cell Ec, and that in the weak link Ew'. Etot = Ep + Ec + Ew The spatial extent of the superfluid weak link 208 extends to a few coherence lengths on either side of the link. For a cell with dimensions much larger than ^o, the weak link 208 is spatially separated from the other elements of the device viz., the displaceable surface 210 and the bulk fluid 204A in the cell; it can be treated as a nonlinear inductor element. For plate displacements that are much smaller than ^o, the superfluid condensate 204A in the cell is robust and the plate is spatially separated and independent of the weak link 208; it may be considered a capacitive element In the device. The total energy may therefore be evaluated using a lumped element approximation. The energy of the fluid in the cell, Ec, is given by thermodynamics In the hydrodynamic regime, dEf = —PdV + ——dMK + TdS — VdP + ——du c 2ms s 2ms where the flow of Cooper pairs of mass 2ms gives rise to a change in volume dV at pressure P; and changes in pressure dP due to the compressibility of the fluid at volume V give rise to changes In the chemical potential dp. Temperature variations In the cell are negligible, so using equation (7): dEc = 0 Ew is the energy stored in the phase shift across the weak link. A Josephson current, I is induced In response to a phase shift according to equation (9). In other words, the induced current is in response to a change in chemical potential (analogous to potential difference or voltage in electrical circuits). The inductance of the weak link is associated with a stored energy AP Ew = I dt— I -'O Ps where the integrand is the mechanical power applied asy- = ^- (analogous to the electrical power which is the product of the voltage / potentlal difference and the current). Using equation (9), this becomes: h d(p 2ms dt h = — Ic cos < / > assuming a sinusoidal Josephson relation. In the linear regime, the displaceable surface 210 is a simple harmonic oscillator with energy: 1 Ps^A2 •■> — kx = 2 Qkms Using the lumped-element approximation and drawing an analogy to the case of a superconducting Cooper Pair Box(CPB), Ew is an inductive term and Ep may be treated as a capacitive term in the CPB circuit Lagrangian. Based on the analysis above, and following the Lagrangian for the CPB, the Lagrangian for the device 200 Is pshzA2 . h L = —;—5-62 + -—Ic cos <b. 8km] * 2ms c This Lagrangian has two explicit canonical degrees of freedom, 0 and ¢. To derive a quantum description of the circuit element, the Lagrangian can be converted to a Hamiltonian, which is then quantised. To assist with this, the circulation of the superfluid is used. The circulation of a fluid is defined as a line integral around a closed contour: x = i> v ■ dl Je where v is the local fluid velocity. For an isotropic superflow In a superfluid, v = Since $ Is well-defined at each point on the contour, it follows that the circulation can only change In multiples of 2n. Therefore: h f h = -—<p V<b-dl = nHn for n e 0,+1,+2 .... 2ms Je Where x0 = —. Thus, the circulation is quantised in units of x0. The generalised circulation and its time derivative, % and % respectively, can be identified as corresponding degrees of freedom to 0 and ¢. These are analogous to the generalized flux and charge, respectively, in the electrical case. The Hamiltonian can be derived from the Lagrangian using: = Pp — £ Where p is the canonical momentum, defined as: dL P =--r dtp The variable <p and its conjugate p are then promoted to operators which obey commutation relations [ip ,p] = ih, and the Hamiltonian function is a function of the operators $ and p. For the superfluid device, the phase can be rewritten in terms of the circulation quantum, x0, using: d<p 2msAP 2nAP dt ~ ft ps ft “ x0 ps This can be used to define: 2tt 2tt — X(t) =--dt ----- and x0 K0 Jo Ps dX _ AP dt ps The Lagrangian can then be rewritten in terms of the generalised circulation as: hlcn2 X2 msa>p x02 + -—cos 2ms The canonical momentum, Q, associated with the dynamical variable X is given by: dL _ 2hlcn2 X dX msojp Xg where Q has units of mass. Converting this to a dimensionless number, n, gives by dividing by 2ms gives a Hamiltonian of: hm^o)2 „ ft = ------E. --i COS A 4 2ms c Analogous to the electrical case, Q and $ can be promoted to operators such that [0 ,Q] = ih. n can be interpreted as the number of (excess) Cooper pairs in the cell. This Hamiltonian is analogous to the Cooper-pair box (CPB) Hamiltonian, XCPB: Mcpb = 4Ecn2 — Ej cos 0 where Ec is the charging energy and Ej is the Josephson energy. In the superfluid device case, these can be equated with: hm„aj2 km2 ft p — a 5 . p —____j c 4IC 2p2A2 ’ J 2ms c Using the CPB analogy, the plasma frequency ho)p = ^QEcEj which is satisfied for aP in the superfluid case. For small ¢, the Hamiltonian is the Hamiltonian for a quantum harmonic oscillator with the second term being the potential energy (| k’^2) and the first term being the kinetic energy (|m'02). It is easy to see that the angular frequency of this oscillator is = <ap. The mapping between the CPB Hamiltonian and the superfluid device Hamiltonian allows analogous physical quantities to be identified between superconducting quantum circuits and the superfluid quantum circuits described herein: Physical quantity Superconducting circuit Device 200 Chemical potential difference V (voltage) 2h ---AP Ps^O Current Electrical current Mass current Generalised flux Magnetic flux Circulation flux Generalised charge Qe=2e Q=2ms Flux quantum h = 2e h X° 2ms Conjugate variables oc 4») (K,Q<xK) Number operator & 2e Q 2m s Ec e2 —, where C= conventional 2C capacitance in the circuit kmi 2pM2 Ej h ___ / e 2e‘c h ------A' 2ms The derivation above neglected the possibility of a constant external pressure difference between maintained between the two sides of the displaceable surface, i.e., different pressures between the inside (Pin) and outside (Pout) of the cell. Such a pressure difference could be implemented by pressurizing through the side walls of the cell or by driving the plate itself. Under such a pressure difference , the equilibrium position of the displaceable surface, xo, will change. The overall displacement of the displaceable surface will then be: Ax(t) = x(t) — x0 It is assumed that a sufficiently low pressure bias between the inside and outside of the cell is maintained such that Ap « A, where A is the superfluid gap, so as to preserve the equilibrium Fermi-Dirac thermal distribution. In this case, the motion of the plate, which continues to be in the simple harmonic regime, is given by: A — &P + J — Pin Pout / 1 Where AP is the pressure difference associated with Josephson tunnelling through the weak link and Po is the constant pressure difference between the inside and outside of the cell. From this, it can be derived that: 2m,AP 2m. / kx \ This changes the capacitive energy stored by the displaceable surface: „ _ 1, 2 _ PsA2fi2 (: 2ms \2 _ lcn / Ep — kx — 9 I r j- * 0 ) — 7 I ) 2 8km$ \ psn / ' Ps' Following the derivation for the non-pressured case results in an updated canonical momentum of: 2Icn ( . Po\ Qa = 777 - 7) = 2ms(n "^O^p ' Ps' Where ng is defined as n Ic^Pq = PsA2P0 8 “ mspsx0^ 2kms The Hamiltonian of the circuit can then be written in the form: Z A2 M = Ec(n — ng) —Ejcostf: From which it can be seen that applying a constant pressure across the cell is equivalent to gating by a constant voltage in a transmon circuit. At low temperatures and gating, the lowest two energy states of this system (which correspond to n=0 and n = l) can be used as a two-level system to implement a qubit. It should be noted that the "height" of the cell (i.e., the dimension perpendicular to the displaceable surface) does not appear in the Hamiltonian. This is a free parameter as such in the design thus far. In some examples, to avoid confinement effects, this dimension is larger than ~ 10pm, e.g., much larger than 10pm. To avoid textural dissipation effects, in some examples this dimension is smaller than textural healing length of the superfluid. FIG. 3A shows an example of a circuit diagram for a superfluid quantum circuit component corresponding to the component described in relation to FIG. 1. The circuit diagram illustrates an analogy to the electrical case. In the example shown, the circuit diagram Is an equivalent circuit for a closed device with superfluid inside the cell and on both sides of the weak link. X denotes the superfluid weak link, and the capacitor is a mechanical fluidic capacitor represented by the displaceable surface of the device. FIG. 3B shows an example of a circuit diagram for a superfluid quantum circuit component corresponding to the component described in relation to FIG. 1. In the example shown, the circuit diagram is an equivalent circuit for a device immersed in a reservoir of superfluid. X denotes the superfluid weak link, and the capacitor is a mechanical fluidic capacitor represented by the displaceable surface of the device. FIG. 4 shows a flow diagram of an example method 400 for operating the quantum circuit devices described herein as part of a quantum computation. For convenience, the method 400 method is described as being performed by a system that implements the quantum computation, e.g., the quantum computer described in relation FIG. 5. At operation 402, the system cools the quantum circuit component to an operating temperature below the superfluid critical temperature. This causes the fluid in the quantum circuit component (e.g., 3He), both inside and outside of the cell, to transition to the superfluid phase. In general, the order parameter of the superfluid samples inside and outside of the cell will have a different phase associated with them, leading to a Josephson effect across the weak link in the cell. In some examples, the operating temperature is a temperature corresponds to a frequency that is less than the plasma frequency divided by two pi. For example, the operating temperature may be less than lmK, e.g., less than 0.5mK, such as 0.4mK. At such temperatures, the difference in energy levels of the quantum circuit are easily resolvable. Such temperatures have been obtained in superfluid 3He experiments. At operation 404, the system encodes a qubit state using two energy states / levels of the quantum circuit component. For example, the lowest two energy state of the quantum circuit component can be used as computational basis states, e.g., the n=0 state of the quantum circuit component can be used as the |0> computational state and the n=l state of the quantum circuit component can be used as the |1> computational state, where n is the number of excess Cooper pairs in the cell of the device. In general, the quantum circuit component can be prepared in a superposition of the |0> and |1> states. Examples of methods for controlling the quantum circuit component to prepare it in a particular state are described with respect to FIG. 1. At operation 406, the system performs a quantum computation. The quantum computation may comprise applying one or more quantum gates to a set of qubits In the system, where at least one of the qubits is implemented by the quantum circuit device. In some examples, a plurality of qubits in the set of qubits are implemented using the quantum circuit components described herein, e.g., each physical qubit in the set of qubits is implemented using a respective superfluid quantum circuit component. In some examples, one or more of the qubits in the set of qubits are implemented using a different qubit scheme, e.g., are superconducting qubits, Ion trap qubits, nuclear qubits or the like. The one or more quantum gates may comprise single qubit quantum gates that act on a single qubit in the plurality of qubits, e.g., Pauli gates, Hadamard gates, or the like. The one or more quantum gates may comprise two-qubit quantum gates that act between two qubits, e.g., CNOT gates, phase shift gates, swap gates, or the like. Higher order multi-qubit gates, such as Toffoli gates, may also be used. At operation 408, the system measures the state of the quantum circuit component after at least a part of the quantum computation has been performed. The measurement may, for example, be a measurement of the energy of the quantum circuit component. The measurement may be used to determine a corresponding state of the quantum circuit component, i.e., whether the measurement corresponds to the |0> state or the |1> state. The resulting measurement and / or state may be used to determine a result of the quantum computation. FIG. 5 shows a schematic overview of an example quantum computer 500. In the example shown, the quantum computer 500 comprises a set of quantum hardware 502 and a set of classical hardware 504 communicatively coupled to the quantum hardware 502. The quantum hardware 502 comprises a set of qubits 506 (e.g., a plurality of qubits) that each behave as a two-level system whose levels representing logical values of 0 and 1. At least one of the qubits 506 (e.g., a plurality of the qubits 506) is implemented using any of the superfluid quantum circuit components described herein. The quantum hardware 502 further comprises one or more control devices 508 for controlling properties of the qubits 506 and / or measuring properties of the qubits 506. In some examples, the control devices 508 may be configured to control couplings between qubits 506, e.g., coupling between adjacent qubits 506 arranged on a grid. The one or more control devices 508 may, for example, control one or more of the qubits 506 to implement one or more quantum gates. Such quantum gates may include one-qubit gate operations such Pauli gates and / or Hadamard gates; two-qubit gate gates such as CX, CZ, and SWAP gates; and gate operations involving three or more qubits, such as Toffoli gates. The classical hardware 504 may comprise one or more classical processors 510 and a classical memory 512. The classical hardware 504 is configured to generate control signals 514 for controlling the quantum hardware 502 that cause the quantum hardware 502 to perform a quantum algorithm. The classical hardware 504 may further be configured to receive qubit measurement results 516 from the quantum hardware 502, and process the measurement results to determine one or more results of the quantum algorithm. The quantum computer 500 can implement a quantum algorithm by initializing the qubits in a selected initial state and then applying a sequence of quantum gate operations to the qubits. The final states of the qubits 506 after the sequence of gates can be measured using the control devices 508. Any system feature as described herein may also be provided as a method feature, and vice versa. As used herein, means plus function features may be expressed alternatively in terms of their corresponding structure. In particular, method aspects may be applied to system aspects, and vice versa. Furthermore, any, some and / or all features in one aspect can be applied to any, some and / or all features in any other aspect, in any appropriate combination. It should also be appreciated that particular combinations of the various features described and defined in any aspects of the invention can be implemented and / or supplied and / or used independently. Although several embodiments have been shown and described, it would be appreciated by those skilled in the art that changes may be made in these embodiments without departing from the principles of this disclosure, the scope of which is defined in the claims.

Claims

1. A quantum circuit component comprising:a reservoir containing a first sample of a fluid, wherein the fluid is in a superfluid phase when cooled below a superfluid critical temperature;a cell within the reservoir comprising a second sample of the fluid, wherein the second sample of fluid is in the same superfluid phase as the first sample of fluid when cooled below the superfluid critical temperature, the cell comprising:one or more weak links between the second sample of the fluid and the first sample of fluid; andone or more elastic and / or displaceable mechanical objects coupled to the one or more weak links via the second sample of fluid,wherein the cell is associated with a characteristic frequency that depends on properties of the one or more elastic and / or displaceable mechanical objects, properties of the one or more weak links and properties of the first and second samples of fluid, wherein the characteristic frequency is associated with quantised nonlinear oscillatory dynamics of the superfluid, andwherein the cell is configured such that the characteristic frequency associated with is less than two pi times a magnitude of a superfluid gap associated with the superfluid phase of the fluid.

2. The quantum circuit component of claim, further comprising an electrode configured to measure one or more properties of the cell indicative of a quantum state of the cell and / or to manipulate one or more properties of the cell to control the quantum state of the cell.

3. The quantum circuit component of claim 2, wherein the electrode comprises a superconducting quantum interference device, SQUID, a superconducting qubit, or a superconducting resonator.

4. The quantum circuit component of any preceding claim, wherein the cell further comprises one or more fixed walls comprising the one or more weak links.

5. The quantum circuit component of any preceding claim, wherein the one or more weak links comprise one or more apertures with dimensions comparable to a coherence length of the fluid when in the superfluid phase.

6. The quantum circuit of claim 5, wherein the one or more weak links comprises an array of N>1 weak links, wherein the weak links in the array are separated by a distance that is greater than the coherence length of the fluid when in the superfluid phase.

7. The quantum circuit of claim 6, wherein the weak links in the array of weak links are configured to function coherently as a single Josephson junction.

8. The quantum circuit component of any preceding claim, wherein the cell dimensions are less than a superfluid texture healing length of the fluid when in the superfluid phase.

9. The quantum circuit component of any preceding claim, wherein, when in the superfluid phase, the first and second samples of fluid have a homogenous uniform texture across the weak links that connect the first and second samples of fluid.

10. The quantum circuit component of any preceding claim, wherein the characteristic frequency, mp, is given by:2msklc p plA2hwhere ms is a mass of quasiparticles of the fluid, ps is a fluid density of the fluid, k is a spring constant associated with the one or more displaceable surfaces, A is an area of the one or more elastic and / or displaceable mechanical components and lc is a critical current for the Josephson effect between the first sample of fluid and the second sample of fluid when in the superfluid phase.

11. The quantum circuit component of any preceding claim, wherein the quantum circuit component can be described below the superfluid critical temperature by a Hamiltonian, M, of a form comprising the following terms:Jf = Ec[n — rig) — Ej cos (f>where Ec is a charging energy associated with the cell, Ej Is a Josephson energy, n Is a number of Cooper pairs in the cell, ng is a number offset caused by a constant pressure difference between the first fluid sample and second fluid sample, and 0 is a superfluld phase difference between the superfluid inside the cell and the superfluid outside the cell.

12. The quantum circuit component of any preceding claim, wherein the fluid is helium-3, and wherein the fluid Is in the B-phase of superfluid helium-3 when cooled below the critical temperature.

13. A method of operating the quantum circuit component of any preceding claim, the method comprising:cooling the quantum circuit component to an operating temperature that is below the superfluid critical temperature;encoding a qubit state in the quantum circuit component using two energy states of the quantum circuit component;performing a quantum computation, comprising applying one or more quantum gates to a set of qubits that includes the quantum circuit component; andmeasuring, subsequent to the quantum computation, the state of the quantum circuit component.

14. The method of claim 13, wherein the operating temperature corresponds to a frequency that is less than the plasma frequency divided by two pi.

15. A quantum circuit comprising one or more quantum circuit components according to any of claims 1 to 12.

16. A quantum computer comprising a plurality of physical qubits, wherein one or more of the plurality of physical qubits comprises a quantum circuit component according to any of claims 1 to 12.25

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