System and methods for producing magic states for universal quantum computation using GKP error correction

Abstract

Applying Gottesman-Kitaev-Preskill (GKP) error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.

Claims

1. A method of producing a magic state for a quantum computer, the method comprising steps of: providing a Gaussian quantum computer including a circuit; generating a Gaussian input state of the circuit; applying a bosonic error correction code to the Gaussian input state, wherein the bosonic error correction code is a Gottesman-Kitaev-Preskill (GKP) error correction code consisting of Gaussian operations and a GKP Pauli eigenstate, the GKP error correction code requiring no non-Gaussian resources beyond the GKP Pauli eigenstate and requiring no additional external qubits, and producing the magic state.

2. The method of claim 1, wherein the Gaussian input state is a thermal state.

3. The method of claim 1, wherein the Gaussian input state is a vacuum state.

4. The method of claim 1, wherein the magic state produced is used with fault tolerant quantum computing.

5. The method of claim 1, wherein the magic state is a H-type magic state.

6. The method of claim 1, wherein the magic state is a T-type magic state.

7. A method for producing a magic state for a quantum computer, the method comprising steps of: providing a Gaussian quantum computer including a circuit; generating a Gaussian input state of the circuit; applying a Gottesman-Kitaev-Preskill (GKP) error correction code to the Gaussian input state, wherein the GKP error correction code comprises both a non-Gaussian state and a Gaussian operation, the GKP error correction code requiring no non-Gaussian resources beyond a GKP Pauli eigenstate and requiring no additional external qubits; and producing the magic state.

8. The method of claim 7, wherein the non-Gaussian state is a GKP logical-0 state.

9. The method of claim 7, wherein the Gaussian input state is a thermal state or a vacuum state.

10. The method of claim 7, wherein the magic state produced is used with fault tolerant quantum computing.

11. The method of claim 7, wherein the magic state is a H-type magic state or a T-type magic state.

12. The method of claim 1, wherein the GKP Pauli eigenstate is a logical-0 state.

13. The method of claim 7, wherein the non-Gaussian state is a GKP Pauli eigenstate.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The following drawings form part of the specification and are included to further demonstrate certain embodiments or various aspects of the invention. In some instances, embodiments of the invention can be best understood by referring to the accompanying drawings in combination with the presented detailed description. The description and accompanying drawings may highlight a certain specific example, or a certain aspect of the invention. However, one skilled in the art will understand that portions of the example or aspect may be used in combination with other examples or aspects of the invention.

(2) FIG. 1A illustrates a circuit used for correction according to an embodiment of the invention.

(3) FIG. 1B illustrates a block diagram of Wigner-function representation of the square-lattice GKP Pauli eigenstates according to an embodiment of the invention.

(4) FIG. 1C illustrates a block diagram of Wigner-function representation of the square-lattice GKP logical Pauli operators in a single unit cell of phase space according to an embodiment of the invention.

(5) FIG. 2 illustrates a graph of GKP error correction of a vacuum according to an embodiment of the invention.

(6) FIG. 3 illustrates a GKP Bloch sphere diagram of representative conditional Bloch vectors for outcomes A-D shown in FIG. 2 according to an embodiment of the invention.

(7) FIG. 4 illustrates a plot diagram of the probability of producing an H-type resource state according to an embodiment of the invention.

(8) FIG. 5 illustrates a graph of square-lattice GKP logical states mapped to equivalent hexagonal-lattice GKP logical states according to an embodiment of the invention.

(9) FIG. 6 illustrates a plot diagram of the probability of producing a distillable resource according to an embodiment of the invention.

DETAILED DESCRIPTION OF THE INVENTION

(10) According to the invention, a magic state for logical-Clifford QC (using a particular bosonic code) can be found within Gaussian QC. GKP-Clifford QC and Gaussian QC combine with no additional non-Gaussian resources into fault-tolerant, universal QC. This is straightforward for qubitscombining Clifford QC based on different Pauli frames can be achieved since stabilizer states of one are magic states for the other.

(11) The following shows how to produce a distillable GKP magic state using GKP error correction on a thermal state (vacuum or finite temperature), along with a complete analysis of the success probability of preparing a high-quality magic state from any given thermal state. The invention is applicable to both square- and hexagonal-lattice GKP encodings.

(12) Notation and conventions used throughout the specification are now defined. Position is defined as

(13) $\hat{p}:=\frac{-i}{\sqrt{2}}\left(\hat{a}-{\hat{a}}^{}\right)$
and momentum is defined as

(14) $\hat{q}:=\frac{-i}{\sqrt{2}}\left(\hat{a}-{\hat{a}}^{}\right)$
for any mode . This means [{circumflex over (q)}; {circumflex over (p)}]=i, with a vacuum variance of in each quadrature and =1.

(15) The Weyl-Heisenberg displacement operators {circumflex over (X)}(s):=e.sup.is{circumflex over (p)} and {circumflex over (Z)}(s):=e.sup.is{circumflex over (q)} displace a state by +s in position and momentum, respectively. For brevity, joint displacement is defined as {circumflex over (V)}(s):={circumflex over (Z)}(s.sub.p){circumflex over (X)}(s.sub.q), where s=(s.sub.q, s.sub.p).sup.T.

(16) The functions (s):=.sub.q<s, > and {circumflex over ()}(s):=.sub.p<s, > denote position-space and momentum-space wave function for a state |>, respectively (tilde indicates momentum space. Any function, including wave functions, can be evaluated with respect to position, ({circumflex over (q)}):=ds (s)|s>.sub.qq<s|, to produce an operator diagonal in the position basis| and similarly for momentum. Finally, Dirac comb with spacing is defined as III.sub.(x):=.sub.n(xn) as a Dirac comb with spacing .

(17) The original square-lattice GKP encoding formulation is described in D. Gottesman, A. Kitaev, and J. Preskill, Encoding a Qubit in an Oscillator, Phys. Rev. A 64, 012310 (2001), otherwise referred to as original formulation. In this original formulation, the wave functions for the logical basis states {|0.sub.L>, |1.sub.L>} are Dirac combs in position space with state-dependent offset: .sub.j,L(s)=III.sub.2{square root over ()}(sj{square root over ()}) for j{0,1}. Their momentum-space wave functions are also Dirac combs but with no offset, different spacing, and a relative phase between the spikes:

(18) ${\stackrel{~}{}}_{J,L}\left(s\right)=\frac{1}{\sqrt{2}}{\left(-1\right)}^{\frac{\mathrm{js}}{\sqrt{}}}{\mathrm{III}}_{\sqrt{}}\left(s\right).$
Note that the momentum-space spikes for |1.sub.L> alternate sign, and those for |0.sub.L> are uniform.

(19) GKP logical operators {circumflex over (X)}.sub.L and {circumflex over (Z)}.sub.L are implemented by displacements {circumflex over (X)}({square root over ()}) and {circumflex over (Z)}({square root over ()}), respectively, while displacements by integer multiples of 2{square root over ()} in either quadrature leave the GKP logical subspace invariant. For later use, the four GKP-encoded logical Pauli is defined as:
.sub.L.sup.:=.sub.jk.sub.jk.sup.|j.sub.L><k.sub.L|Equation (1) where .sub.jk.sup. is the jkth element of Pauli matrix .sup. (with .sup.0=I). Note that .sub.L.sup. have support only on the GKP logical subspace, while {circumflex over (X)}.sub.L and {circumflex over (Z)}.sub.L have full support. Finally, the (rank-two) projector is denoted onto the GKP logical subspace:
{circumflex over ()}.sub.GKP:={circumflex over ()}.sub.L.sup.0={circumflex over ()}.sub.0,L({circumflex over (q)}).sub.0,L({circumflex over (p)})={tilde over ()}.sub.0,L({circumflex over (p)}){tilde over ()}.sub.0,L({circumflex over (q)})Equation (2)

(20) Kraus operator for GKP error correction is now discussed. In its original formulation, GKP error correction is a quantum operation that corrects an encoded qubit that has acquired some noise (leakage of its wave function outside of the logical subspace) by projecting it back into the GKP logical subspace, possibly at the expense of an unintended logical operation. A standard implementation of error correction strives to avoid these unintended logical operations (residual errors).

(21) In what follows, the machinery of error correction is applied to a known Gaussian state, which means the outcome-dependent final state is known perfectly. Since the input state is known (which is usually not true of quantum error correction), exactly what state results from the procedure are known.

(22) GKP error correction proceeds in two steps: First, the one quadrature is corrected, then the conjugate quadrature. The Kraus operator is now defined that corrects just the q quadrature {circumflex over (K)}.sub.EC.sup.q(t) via the circuit (read right to left) as shown in FIG. 1A where the controlled operation is .sub.Z=e.sup.i{circumflex over (q)}.Math.{circumflex over (q)}, and t is the measurement outcome. This circuit differs from the original formulation in that the correction here is a negative displacement by t rather than by t rounded to the nearest integer multiple of {square root over ()}. The outputs may differ by a logical operation {circumflex over (X)}({square root over ()}), but this difference is unimportant because the input state is known.

(23) Direct evaluation shows {circumflex over (K)}.sub.EC.sup.q(t)={tilde over ()}.sub.0,L({circumflex over (q)}){circumflex over (X)}(t). A similar calculation shows that the Kraus operator for correcting the p quadrature is {circumflex over (K)}.sub.EC.sup.p(t)={tilde over ()}.sub.0,L({circumflex over (p)}){circumflex over (Z)}(t). Applying both corrections (in either order since they commute up to a phase) performs full GKP error correction:
{circumflex over (K)}.sub.EC(t)={circumflex over (K)}.sub.EC.sup.p(t.sub.p){circumflex over (K)}.sub.EC.sup.q(t.sub.q)={circumflex over ()}.sub.GKP{circumflex over (V)}(t)Equation (3) with measurement outcomes t=(t.sub.q, t.sub.p).sup.T. This Kraus operator (i) displaces the state by an outcome-dependent amount {circumflex over (V)}(t), and then (ii) projects it back into the GKP logical subspace with {circumflex over ()}.sub.GKP.

(24) Applying {circumflex over (K)}.sub.EC(t) to an input state {circumflex over ()}.sub.in in produces the unnormalized state {circumflex over (p)}(t)={circumflex over (K)}.sub.EC(t){circumflex over ()}.sub.in {circumflex over (K)}.sub.EC.sup.(t), where the bar indicates lack of normalization. The joint probability density function (pdf) for the outcomes, pdf(t)=Tr[{circumflex over (p)}(t)], normalizes the output state: {circumflex over ()}(t)={circumflex over (p)}(t)/pdf(t).

(25) Now, Bloch vector for the error-corrected state is discussed. Using the logical basis in Equation (1) the output state {circumflex over ()} (t)= .sub. r.sub.(t){circumflex over ()}.sub.L.sup. is represented by a 4-component Bloch vector r(t) with outcome-dependent coefficients r.sub.(t):=Tr[{circumflex over ()}(t){circumflex over ()}.sub.L.sup.]. For the unnormalized state, r.sub.0(t)=pdf(t) and for the normalized state, r.sub.0(t)=1. In what follows, the notation r=(r.sub.0, {right arrow over (r)}), where {right arrow over (r)} is the ordinary (3-component) Bloch vector within r.

(26) Wigner functions are employed for the logical basis states of the original formulation as shown in FIG. 1B, to find the Wigner functions for the GKP-encoded Pauli operators and the GKP logical identity, Equation (1). Their explicit form is

(27) $\begin{array}{cc}{W}_{{}_L^{}}\left(x\right)={.Math.}_{n{}^2}\frac{{\left(-1\right)}^{n.Math.{\stackrel{\_}{}}_{}}}{2}{}^{\left(2\right)}\left[x-\left(n+\frac{{}_{}}{2}\right)\sqrt{}\right]& \mathrm{Equation}\left(4\right)\end{array}$ where x=(q, p).sup.T, l.sub.0(0,0).sup.T, l.sub.1(1,0).sup.T, l.sub.2(1,1).sup.T, l.sub.3(0,1).sup.T, and l.sub. is l.sub. with its entries swapped. FIG. 1C illustrates a block diagram of Wigner-function representation of the square-lattice GKP logical Pauli operators in a single unit cell of phase space with dimensions (2{square root over ()})(2{square root over ()}). The states are normalized to 1 over one unit cell which determines the coefficients c.

(28) Since {circumflex over ()}.sub.GKP{circumflex over ()}.sub.L.sup.{circumflex over ()}.sub.GKP={circumflex over ()}.sub.L.sup. the projection using {circumflex over ()}.sub.GKP is skipped and the unnormalized Bloch vector components are directly calculated from the overlap of the unnormalized error-corrected state {circumflex over (p)}(t) with the logical Paulis. The overlaps in the Wigner representation are found:
r.sub.(t)=Tr[{circumflex over (p)}(t){circumflex over ()}.sub.L.sup.]=Tr[{circumflex over (V)}(t){circumflex over ()}.sub.in{circumflex over (V)}.sup.(t){circumflex over ()}.sub.]=2d.sup.2W.sub.in(x+t)W.sub..sub.L.sub.(x)Equation (5), where W.sub.in(x) is the Wigner function of the input state {circumflex over ()}.sub.in. Note that r.sub.0(t)=Tr[{circumflex over (p)}(t)]=pdf(t), which is normalized over a unit cell of size (2{square root over ()})(2{square root over ()}) (since the full pdf is periodic). The normalized Bloch 4-vector is r(t):=r(t)/{circumflex over (r)}.sub.0(t).

(29) In what follows, GKP error correction is applied to a general Gaussian state, i.e., an input state whose Wigner function is W.sub.in(x)=G.sub.x.sub.0.sub.(x), where G.sub.x.sub.0.sub.(x) is a normalized Gaussian with mean vector x.sub.0 and covariance matrix .

(30) Equation (5) can be evaluated analytically when the input state is Gaussian:

(31) $\begin{array}{cc}{\stackrel{\_}{r}}_{}\left(t\right)={\frac{1}{4}\left[G_{0,{\left(4.Math.\right)}^{-1}}\left(v\right)\right]}^{-1}\left(v+\frac{{\stackrel{\_}{}}_{}}{2},\right)& \mathrm{Equation}\left(6\right)\end{array}$
where

(32) $=\frac{1}{2}{.Math.}^{-1},v=\left[\frac{1}{2}{}_{}-\frac{1}{\sqrt{}}\left(x_o+t\right)\right],$
and the Riemann (a.k.a. Siegel) theta function is defined as (z, ):=.sub.m.sub.n exp[2i( m.sup.Tm+m.sup.Tz] for .sub.n. The set .sub.n denotes the Siegel upper half space|, i.e., the set of all complex, symmetric, nn matrices with positive definite imaginary part. The overall coefficient

(33) $\frac{1}{4}$
ensures that pdf(t) is normalized over a single unit cell.

(34) GKP magic states from error correction is now discussed. GKP error correction of a Gaussian state yields a known, random state encoded in the GKP logical subspace. Unless that state is highly mixed or too close to a logical Pauli eigenstate, it can be used as a (noisy) magic state to elevate GKP Clifford QC to fault-tolerant universal QC. It has been suggested to couple a vacuum mode to an external qubit to perform GKP error correction and then post-select an outcome close to t0 to produce a logical H-type state. In fact, neither post-selection nor interaction with a material qubit is required.

(35) With access to a supply of |0.sub.L> states, there is no need for any other resources except Gaussian QC, and nearly any outcome t from applying GKP error correction to the vacuum state produces a distillable H-type magic state as shown in FIG. 2. This is because there are 12 H-type magic states (all related by Cliffords to |+H.sub.L>), and any of them will do the job. The relevant quantity is the fidelity F to the closest H-type state. Without loss of generality, assume this is |+H.sub.L>, whose Bloch 3-vector is {right arrow over (r)}.sub.H=(1, 0, 1). If not, GKP Cliffords may be applied until it is. Then, F=<(+H.sub.L|{circumflex over ()}(t)|+H.sub.L>=[1+{right arrow over (r)}.sub.H.Math.{right arrow over (r)}(t)]. As shown in FIG. 2, the only outcomes that do not yield a distillable magic state are marked with a white x (these yield GKP Pauli eigenstates). Representative conditional Bloch vectors for outcomes (A-D) are shown on the GKP Bloch sphere in FIG. 3.

(36) Purity is not required either. Applying GKP error correction to a thermal state also succeeds with nonzero probability, and distillation is possible, so as long as n<n.sub.thresh,H=0.366. FIG. 4 illustrates the probability of producing an H-type resource state of at least fidelity F by performing GKP error correction on a thermal state of mean occupation n. Resource states with fidelity higher than the distillation threshold (F >0.853) can be distilled into higher-quality|+H.sub.L> states. A thermal state is Gaussian with x.sub.0=0 and =(n+)I, which is plugged into Equation (6) to produce the plot of FIG. 4. Since thermal states are biased towards magic states in the xz-plane of the Bloch sphere shown in FIG. 3, maximum fidelity with those magic states in the xy- and yz-plane drops below the distillation threshold first as n increases, leading to the kinks in FIG. 4.

(37) Most high-purity, Gaussian states can be GKP-error corrected into a distillable magic state because most states do not preferentially error correct to a Pauli eigenstate. For the vacuum, pdf(t) is always between 0.066 and 0.094, i.e., all outcomes, and thus a wide variety of states, are roughly equally likely.

(38) The results can be extended to the hexagonal-lattice GKP code by simply modifying the Gaussian state to be error corrected as follows. With .Math. defined as the Gaussian unitary such that .Math.|{circumflex over ()}.sub.L.sup.square)=|{circumflex over ()}.sub.L.sup.hex), where the logical state is the same although the encoding differs. Let {circumflex over ()} be a Gaussian state to be GKP error corrected using the hexagonal lattice, with x.sub.0=0 and covariance . Then, the equivalent state to be GKP error corrected using the square lattice is {circumflex over ()}.sub.in=.Math..sup.{circumflex over ()}.sub.in.Math., which is Gaussian with x.sub.0=0 and covariance =S.sup.1S.sup.T, where

(39) $S={\left(2\sqrt{3}\right)}^{-\frac{1}{2}}\left(\begin{array}{cc}2& -1\\ 0& \sqrt{3}\end{array}\right).$
This mapping is shown for {circumflex over ()}.sub.in=|vac><vac| in FIG. 5. More specifically, FIG. 5 illustrates a graph of square-lattice GKP logical states mapped to equivalent hexagonal-lattice GKP logical states according to an embodiment of the invention. A vacuum state on the hexagonal lattice (1 error ellipse) is mapped to a squeezed state on the square lattice under .Math..sup..

(40) Using this mapping, results for hexagonal-lattice GKP error correction can be found by reusing Equation (6) with the modified state. Vacuum and thermal states are biased towards the xz-plane of the Bloch sphere in the square-lattice encoding but unbiased with respect to all three Pauli axes in the hexagonal-lattice encoding. Thus, in FIG. 6, the fidelity of hexagonal-lattice GKP error correction of a thermal state of mean occupation n is plotted against T-type magic states such as |+T.sub.L.sup.hex>, which has Bloch 3-vector

(41) ${\stackrel{.fwdarw.}{r}}_T=\frac{1}{\sqrt{3}}\left(1,1,1\right).$
Resource states whose Bloch vectors lie on or within the stabilizer octahedron (F0.789) cannot be distilled, which occurs at n.sub.bound,T=0.468. For T states, a tight distillation threshold has been proven for F >0.8273 [28], which occurs for n.sub.thresh,T=0.391.

(42) These results can be generalized straightforwardly to the case of imperfect |0.sub.L> states represented approximately as {circumflex over (K)}.sub.|0.sub.L> where {circumflex over (K)}.sub.:=e.sup.{circumflex over ()}{circumflex over ()} for some <<1 (and ignoring normalization).

(43) The approximate GKP Pauli are {circumflex over (K)}.sub.{circumflex over ()}.sub.L.sup.{circumflex over (K)}.sub.. Noting that Tr({circumflex over ()}{circumflex over (K)}.sub.{circumflex over ()}.sub.L.sup.{circumflex over (K)}.sub.)=Tr({circumflex over (K)}.sub.{circumflex over ()}{circumflex over (K)}.sub..sub.L.sup.) so the (Gaussian) imperfections represented by {circumflex over (K)}.sub. can be accounted for by applying them to the input state instead. Since the fidelity requirements for magic-state distillation are orders of magnitude less than those for fault-tolerant Clifford QC, any residual noise introduced by {circumflex over (K)}.sub. will not qualitatively change the main result.

(44) An alternate embodiment of the invention is to perform error correction as heterodyne detection (measurement in the coherent-state basis) on half of a GKP encoded Bell pair. Noting that a Bell state can be written (ignoring normalization) as .sub.{circumflex over ()}.sub.L.sup..Math.{circumflex over ()}.sub.L.sup.. Then, a coherent-state measurement on the first mode with outcome produces .sub. Tr(|><|.sub.L.sup.).sub.L.sup. on the second mode, which agrees with Equation (5) using {circumflex over ()}.sub.in as vacuum and t={square root over (2)}(Re, Im).sup.T. Intuitively, this is just Knill-type error correction, which involves teleporting the state to be corrected through an encoded Bell pair and reinterpreting vacuum teleportation as heterodyne detection.

(45) Therefore, GKP-Clifford QC and Gaussian QC combine with no additional non-Gaussian resources into fault-tolerant, universal QC. This is because |vac> is a distillable magic state that elevates GKP Clifford QC to fault-tolerant universality. Practically, this means there is no longer any need for the pursuit of creating cubic phase states if GKP encoding is used. By focusing on making high-quality GKP |0.sub.L> states, the rest is all Gaussian.

(46) Fundamentally, this shows that two efficiently simulable subtheories, when used together, are universal and fault tolerant. This is straightforward for qubits: just combine Clifford QC based on different Pauli frames. For example, {{circumflex over (X)}, , {circumflex over (Z)}} and {, , {circumflex over (Z)}{circumflex over (Z)}} since stabilizer states of one are magic states for the other. The invention demonstrates the magic of error correction by deploying it in a nonstandard way to produce resource states from a known, easy-to-prepare state. The wilderness space outside a bosonic code's logical subspace may be rich in other resources, too, for example, providing the means to produce other logical states or perform logical operations more easily than would be possible by restricting to the logical subspace. It is contemplated that the invention is likely to extend beyond GKP to other bosonic codes such as rotation-symmetric codes, experimentally proven cat codes, and multi-mode GKP codes.

(47) While the disclosure is susceptible to various modifications and alternative forms, specific exemplary embodiments of the invention have been shown by way of example in the drawings and have been described in detail. It should be understood, however, that there is no intent to limit the disclosure to the particular embodiments disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the scope of the disclosure as defined by the appended claims.